unit exponents and scientific notation homework 5 answer key

Worksheet and Answer Key

• Scientific Notation
• Operations with Scientific Notation

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Chapter 6: Polynomials

6.3 Scientific Notation (Homework Assignment)

Scientific notation is a convenient notation system used to represent large and small numbers. Examples of these are the mass of the sun or the mass of an electron in kilograms. Simplifying basic operations such as multiplication and division with these numbers requires using exponential properties.

Scientific notation has two parts: a number between one and nine and a power of ten, by which that number is multiplied.

[latex]\text{Scientific notation: }a \times 10^b, \text{ where }1 \le a \le 9[/latex]

The exponent tells how many times to multiply by 10. Each multiple of 10 shifts the decimal point one place. To decide which direction to move the decimal (left or right), recall that positive exponents means there is big number (larger than ten) and negative exponents means there is a small number (less than one).

Example 6.3.1

Convert 14,200 to scientific notation.

[latex]\begin{array}{rl} 1.42&\text{Put a decimal after the first nonzero number} \\ \times 10^4 & \text{The exponent is how many times the decimal moved} \\ 1.42 \times 10^4& \text{Combine to yield the solution} \end{array}[/latex]

Example 6.3.2

Convert 0.0028 to scientific notation.

[latex]\begin{array}{rl} 2.8&\text{Put a decimal after the first nonzero number} \\ \times 10^{-3}&\text{The exponent is how many times the decimal moved} \\ 2.8\times 10^{-3}&\text{Combine to yield the solution} \end{array}[/latex]

Example 6.3.3

Convert 3.21 × 10 5 to standard notation.

Starting with 3.21, Shift the decimal 5 places to the right, or multiply 3.21 by 10 5 .

321,000 is the solution.

Example 6.3.4

Convert 7.4 × 10 −3 to standard notation

Shift the decimal 3 places to the left, or divide 6.4 by 10 3 .

0.0074 is the solution.

Working with scientific notation is easier than working with other exponential notation, since the base of the exponent is always 10. This means that the exponents can be treated separately from any other numbers. For instance:

Example 6.3.5

Multiply (2.1 × 10 −7 )(3.7 × 10 5 ).

First, multiply the numbers 2.1 and 3.7, which equals 7.77.

Second, use the product rule of exponents to simplify the expression 10 −7 × 10 5 , which yields 10 −2 .

Combine these terms to yield the solution 7.77 × 10 −2 .

Example 6.3.6

(4.96 × 10 4 ) ÷ (3.1 × 10 −3 )

First, divide: 4.96 ÷ 3.1 = 1.6

Second, subtract the exponents (it is a division): 10 4− −3 = 10 4 + 3 = 10 7

Combine these to yield the solution 1.6 × 10 7 .

For questions 1 to 6, write each number in scientific notation.

For questions 7 to 12, write each number in standard notation.

• 2.56 × 10 2

For questions 13 to 20, simplify each expression and write each answer in scientific notation.

• (7 × 10 −1 )(2 × 10 −3 )
• (2 × 10 −6 )(8.8 × 10 −5 )
• (5.26 × 10 −5 )(3.16 × 10 −2 )
• (5.1 × 10 6 )(9.84 × 10 −1 )
• [latex]\dfrac{(2.6 \times 10^{-2})(6 \times 10^{-2})}{(4.9 \times 10^1)(2.7 \times 10^{-3})}[/latex]
• [latex]\dfrac{(7.4 \times 10^4)(1.7 \times 10^{-4})}{(7.2 \times 10^{-1})(7.32 \times 10^{-1})}[/latex]
• [latex]\dfrac{(5.33 \times 10^{-6})(9.62 \times 10^{-2})}{(5.5 \times 10^{-5})^2}[/latex]
• [latex]\dfrac{(3.2 \times 10^{-3})(5.02 \times 10^0)}{(9.6 \times 10^3)^{-4}}[/latex]

Answer Key 6.3

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Chapter 1 Prerequisites

1.2 Exponents and Scientific Notation

Learning objectives.

In this section, students will:

• Use the product rule of exponents.
• Use the quotient rule of exponents.
• Use the power rule of exponents.
• Use the zero exponent rule of exponents.
• Use the negative rule of exponents.
• Find the power of a product and a quotient.
• Simplify exponential expressions.
• Use scientific notation.

Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per frame and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number.

Using a calculator, we enter[latex]\,2,048\,×\,1,536\,×\,48\,×\,24\,×\,3,600\,[/latex]and press ENTER. The calculator displays 1.304596316E13. What does this mean? The “E13” portion of the result represents the exponent 13 of ten, so there are a maximum of approximately[latex]\,1.3\,×\,{10}^{13}\,[/latex]bits of data in that one-hour film. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers.

Using the Product Rule of Exponents

Consider the product[latex]\,{x}^{3}\cdot {x}^{4}.\,[/latex]Both terms have the same base, x , but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.

The result is that[latex]\,{x}^{3}\cdot {x}^{4}={x}^{3+4}={x}^{7}.[/latex]

Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents.

Now consider an example with real numbers.

We can always check that this is true by simplifying each exponential expression. We find that[latex]\,{2}^{3}\,[/latex]is 8,[latex]\,{2}^{4}\,[/latex]is 16, and[latex]\,{2}^{7}\,[/latex]is 128. The product[latex]\,8\cdot 16\,[/latex]equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.

The Product Rule of Exponents

For any real number[latex]\,a\,[/latex]and natural numbers[latex]\,m\,[/latex]and[latex]\,n,[/latex] the product rule of exponents states that

Using the Product Rule

Write each of the following products with a single base. Do not simplify further.

• [latex]{t}^{5}\cdot {t}^{3}[/latex]
• [latex]{\left(-3\right)}^{5}\cdot \left(-3\right)[/latex]
• [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}[/latex]

Use the product rule to simplify each expression.

• [latex]{t}^{5}\cdot {t}^{3}={t}^{5+3}={t}^{8}[/latex]
• [latex]{\left(-3\right)}^{5}\cdot \left(-3\right)={\left(-3\right)}^{5}\cdot {\left(-3\right)}^{1}={\left(-3\right)}^{5+1}={\left(-3\right)}^{6}[/latex]

At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.

Notice we get the same result by adding the three exponents in one step.

• [latex]{k}^{6}\cdot {k}^{9}[/latex]
• [latex]{\left(\frac{2}{y}\right)}^{4}\cdot \left(\frac{2}{y}\right)[/latex]
• [latex]{t}^{3}\cdot {t}^{6}\cdot {t}^{5}[/latex]
• [latex]{k}^{15}[/latex]
• [latex]{\left(\frac{2}{y}\right)}^{5}[/latex]
• [latex]{t}^{14}[/latex]

Using the Quotient Rule of Exponents

The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as[latex]\,\frac{{y}^{m}}{{y}^{n}},[/latex] where[latex]\,m>n.\,[/latex]Consider the example[latex]\,\frac{{y}^{9}}{{y}^{5}}.\,[/latex]Perform the division by canceling common factors.

Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.

In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.

For the time being, we must be aware of the condition[latex]\,m>n.\,[/latex]Otherwise, the difference[latex]\,m-n\,[/latex]could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.

The Quotient Rule of Exponents

For any real number[latex]\,a\,[/latex]and natural numbers[latex]\,m\,[/latex]and[latex]\,n,[/latex] such that[latex]\,m>n,[/latex] the quotient rule of exponents states that

Using the Quotient Rule

• [latex]\frac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}[/latex]
• [latex]\frac{{t}^{23}}{{t}^{15}}[/latex]
• [latex]\frac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}[/latex]

Use the quotient rule to simplify each expression.

• [latex]\frac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}={\left(-2\right)}^{14-9}={\left(-2\right)}^{5}[/latex]
• [latex]\frac{{t}^{23}}{{t}^{15}}={t}^{23-15}={t}^{8}[/latex]
• [latex]\frac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}={\left(z\sqrt{2}\right)}^{5-1}={\left(z\sqrt{2}\right)}^{4}[/latex]
• [latex]\frac{{s}^{75}}{{s}^{68}}[/latex]
• [latex]\frac{{\left(-3\right)}^{6}}{-3}[/latex]
• [latex]\frac{{\left(e{f}^{2}\right)}^{5}}{{\left(e{f}^{2}\right)}^{3}}[/latex]
• [latex]{s}^{7}[/latex]
• [latex]{\left(-3\right)}^{5}[/latex]
• [latex]{\left(e{f}^{2}\right)}^{2}[/latex]

Using the Power Rule of Exponents

Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the power rule of exponents . Consider the expression[latex]\,{\left({x}^{2}\right)}^{3}.\,[/latex]The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.

The exponent of the answer is the product of the exponents:[latex]\,{\left({x}^{2}\right)}^{3}={x}^{2\cdot 3}={x}^{6}.\,[/latex]In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.

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Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.

The Power Rule of Exponents

For any real number[latex]\,a\,[/latex]and positive integers[latex]\,m\,[/latex]and[latex]\,n,[/latex]the power rule of exponents states that

Using the Power Rule

• [latex]{\left({x}^{2}\right)}^{7}[/latex]
• [latex]{\left({\left(2t\right)}^{5}\right)}^{3}[/latex]
• [latex]{\left({\left(-3\right)}^{5}\right)}^{11}[/latex]

Use the power rule to simplify each expression.

• [latex]{\left({x}^{2}\right)}^{7}={x}^{2\cdot 7}={x}^{14}[/latex]
• [latex]{\left({\left(2t\right)}^{5}\right)}^{3}={\left(2t\right)}^{5\cdot 3}={\left(2t\right)}^{15}[/latex]
• [latex]{\left({\left(-3\right)}^{5}\right)}^{11}={\left(-3\right)}^{5\cdot 11}={\left(-3\right)}^{55}[/latex]
• [latex]{\left({\left(3y\right)}^{8}\right)}^{3}[/latex]
• [latex]{\left({t}^{5}\right)}^{7}[/latex]
• [latex]{\left({\left(-g\right)}^{4}\right)}^{4}[/latex]
• [latex]{\left(3y\right)}^{24}[/latex]
• [latex]{t}^{35}[/latex]
• [latex]{\left(-g\right)}^{16}[/latex]

Using the Zero Exponent Rule of Exponents

Return to the quotient rule. We made the condition that[latex]\,m>n\,[/latex]so that the difference[latex]\,m-n\,[/latex]would never be zero or negative. What would happen if[latex]\,m=n?[/latex] In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. To see how this is done, let us begin with an example.

If we were to simplify the original expression using the quotient rule, we would have

If we equate the two answers, the result is[latex]\,{t}^{0}=1.\,[/latex]This is true for any nonzero real number, or any variable representing a real number.

The sole exception is the expression[latex]\,{0}^{0}.\,[/latex]This appears later in more advanced courses, but for now, we will consider the value to be undefined.

The Zero Exponent Rule of Exponents

For any nonzero real number[latex]\,a,[/latex] the zero exponent rule of exponents states that

Using the Zero Exponent Rule

Simplify each expression using the zero exponent rule of exponents.

• [latex]\frac{{c}^{3}}{{c}^{3}}[/latex]
• [latex]\frac{-3{x}^{5}}{{x}^{5}}[/latex]
• [latex]\frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}[/latex]
• [latex]\frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}[/latex]

Use the zero exponent and other rules to simplify each expression.

• <[latex]\begin{array}{ccc}\hfill \frac{{c}^{3}}{{c}^{3}}& =& {c}^{3-3}\hfill \\ & =& {c}^{0}\hfill \\ & =& 1\hfill \end{array}[/latex]</li>
• <[latex]\begin{array}{ccc}\hfill \frac{-3{x}^{5}}{{x}^{5}}& =& -3\cdot \frac{{x}^{5}}{{x}^{5}}\hfill \\ & =& -3\cdot {x}^{5-5}\hfill \\ & =& -3\cdot {x}^{0}\hfill \\ & =& -3\cdot 1\hfill \\ & =& -3\hfill \end{array}[/latex]</li>
• <[latex]\begin{array}{cccc}\hfill \frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}& =& \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{1+3}}\hfill & \phantom{\rule{3em}{0ex}}\text{Use the product rule in the denominator}.\hfill \\ & =& \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{4}}\hfill & \phantom{\rule{3em}{0ex}}\text{Simplify}.\hfill \\ & =& {\left({j}^{2}k\right)}^{4-4}\hfill & \phantom{\rule{3em}{0ex}}\text{Use the quotient rule}.\hfill \\ & =& {\left({j}^{2}k\right)}^{0}\hfill & \phantom{\rule{3em}{0ex}}\text{Simplify}.\hfill \\ & =& 1\hfill & \end{array}[/latex]</li>
• <[latex]\begin{array}{cccc}\hfill \frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}& =& 5{\left(r{s}^{2}\right)}^{2-2}\hfill & \phantom{\rule{5em}{0ex}}\text{Use the quotient rule}.\hfill \\ & =& 5{\left(r{s}^{2}\right)}^{0}\hfill & \phantom{\rule{5em}{0ex}}\text{Simplify}.\hfill \\ & =& 5\cdot 1\hfill & \phantom{\rule{5em}{0ex}}\text{Use the zero exponent rule}.\hfill \\ & =& 5\hfill & \phantom{\rule{5em}{0ex}}\text{Simplify}.\hfill \end{array}[/latex]</li>
• [latex]\frac{{t}^{7}}{{t}^{7}}[/latex]
• [latex]\frac{{\left(d{e}^{2}\right)}^{11}}{2{\left(d{e}^{2}\right)}^{11}}[/latex]
• [latex]\frac{{w}^{4}\cdot {w}^{2}}{{w}^{6}}[/latex]
• [latex]\frac{{t}^{3}\cdot {t}^{4}}{{t}^{2}\cdot {t}^{5}}[/latex]
• [latex]1[/latex]
• [latex]\frac{1}{2}[/latex]

Using the Negative Rule of Exponents

Another useful result occurs if we relax the condition that[latex]\,m>n\,[/latex]in the quotient rule even further. For example, can we simplify[latex]\,\frac{{h}^{3}}{{h}^{5}}?\,[/latex]Yes—that is, where the difference [latex]\,m - n\,[/latex]is negative—and we can use the negative rule of exponents to simplify the expression to its reciprocal.

Divide one exponential expression by another with a larger exponent. Use our example,[latex]\,\frac{{h}^{3}}{{h}^{5}}.[/latex]

Putting the answers together, we have[latex]\,{h}^{-2}=\frac{1}{{h}^{2}}.\,[/latex]This is true for any nonzero real number, or any variable representing a nonzero real number.

A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.

We have shown that the exponential expression[latex]\,{a}^{n}\,[/latex]is defined when[latex]\,n\,[/latex]is a natural number, 0, or the negative of a natural number. That means that[latex]\,{a}^{n}\,[/latex]is defined for any integer[latex]\,n.\,[/latex]Also, the product and quotient rules and all of the rules we will look at soon hold for any integer[latex]\,n.[/latex]

The Negative Rule of Exponents

For any nonzero real number[latex]\,a\,[/latex]and natural number[latex]\,n,[/latex] the negative rule of exponents states that

Using the Negative Exponent Rule

Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

• [latex]\frac{{\theta }^{3}}{{\theta }^{10}}[/latex]
• [latex]\frac{{z}^{2}\cdot z}{{z}^{4}}[/latex]
• [latex]\frac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}[/latex]
• [latex]\frac{{\theta }^{3}}{{\theta }^{10}}={\theta }^{3-10}={\theta }^{-7}=\frac{1}{{\theta }^{7}}[/latex]
• [latex]\frac{{z}^{2}\cdot z}{{z}^{4}}=\frac{{z}^{2+1}}{{z}^{4}}=\frac{{z}^{3}}{{z}^{4}}={z}^{3-4}={z}^{-1}=\frac{1}{z}[/latex]
• [latex]\frac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}={\left(-5{t}^{3}\right)}^{4-8}={\left(-5{t}^{3}\right)}^{-4}=\frac{1}{{\left(-5{t}^{3}\right)}^{4}}[/latex]
• [latex]\frac{{\left(-3t\right)}^{2}}{{\left(-3t\right)}^{8}}[/latex]
• [latex]\frac{{f}^{47}}{{f}^{49}\cdot f}[/latex]
• [latex]\frac{2{k}^{4}}{5{k}^{7}}[/latex]
• [latex]\frac{1}{{\left(-3t\right)}^{6}}[/latex]
• [latex]\frac{1}{{f}^{3}}[/latex]
• [latex]\frac{2}{5{k}^{3}}[/latex]

Using the Product and Quotient Rules

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

• [latex]{b}^{2}\cdot {b}^{-8}[/latex]
• [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}[/latex]
• [latex]\frac{-7z}{{\left(-7z\right)}^{5}}[/latex]
• [latex]{b}^{2}\cdot {b}^{-8}={b}^{2-8}={b}^{-6}=\frac{1}{{b}^{6}}[/latex]
• [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}={\left(-x\right)}^{5-5}={\left(-x\right)}^{0}=1[/latex]
• [latex]\frac{-7z}{{\left(-7z\right)}^{5}}=\frac{{\left(-7z\right)}^{1}}{{\left(-7z\right)}^{5}}={\left(-7z\right)}^{1-5}={\left(-7z\right)}^{-4}=\frac{1}{{\left(-7z\right)}^{4}}[/latex]
• [latex]{t}^{-11}\cdot {t}^{6}[/latex]
• [latex]\frac{{25}^{12}}{{25}^{13}}[/latex]
• [latex]{t}^{-5}=\frac{1}{{t}^{5}}[/latex]
• [latex]\frac{1}{25}[/latex]

Finding the Power of a Product

To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider[latex]\,{\left(pq\right)}^{3}.\,[/latex]We begin by using the associative and commutative properties of multiplication to regroup the factors.

In other words,[latex]\,{\left(pq\right)}^{3}={p}^{3}\cdot {q}^{3}.[/latex]

The Power of a Product Rule of Exponents

For any real numbers[latex]\,a\,[/latex]and[latex]\,b\,[/latex]and any integer[latex]\,n,[/latex] the power of a product rule of exponents states that

Using the Power of a Product Rule

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

• [latex]{\left(a{b}^{2}\right)}^{3}[/latex]
• [latex]{\left(2t\right)}^{15}[/latex]
• [latex]{\left(-2{w}^{3}\right)}^{3}[/latex]
• [latex]\frac{1}{{\left(-7z\right)}^{4}}[/latex]
• [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}[/latex]

Use the product and quotient rules and the new definitions to simplify each expression.

• [latex]{\left(a{b}^{2}\right)}^{3}={\left(a\right)}^{3}\cdot {\left({b}^{2}\right)}^{3}={a}^{1\cdot 3}\cdot {b}^{2\cdot 3}={a}^{3}{b}^{6}[/latex]
• [latex]{\left(2t\right)}^{15}={\left(2\right)}^{15}\cdot {\left(t\right)}^{15}={2}^{15}{t}^{15}=32,768{t}^{15}[/latex]
• [latex]{\left(-2{w}^{3}\right)}^{3}={\left(-2\right)}^{3}\cdot {\left({w}^{3}\right)}^{3}=-8\cdot {w}^{3\cdot 3}=-8{w}^{9}[/latex]
• [latex]\frac{1}{{\left(-7z\right)}^{4}}=\frac{1}{{\left(-7\right)}^{4}\cdot {\left(z\right)}^{4}}=\frac{1}{2,401{z}^{4}}[/latex]
• [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}={\left({e}^{-2}\right)}^{7}\cdot {\left({f}^{2}\right)}^{7}={e}^{-2\cdot 7}\cdot {f}^{2\cdot 7}={e}^{-14}{f}^{14}=\frac{{f}^{14}}{{e}^{14}}[/latex]
• [latex]{\left({g}^{2}{h}^{3}\right)}^{5}[/latex]
• [latex]{\left(5t\right)}^{3}[/latex]
• [latex]{\left(-3{y}^{5}\right)}^{3}[/latex]
• [latex]\frac{1}{{\left({a}^{6}{b}^{7}\right)}^{3}}[/latex]
• [latex]{\left({r}^{3}{s}^{-2}\right)}^{4}[/latex]
• [latex]{g}^{10}{h}^{15}[/latex]
• [latex]125{t}^{3}[/latex]
• [latex]-27{y}^{15}[/latex]
• [latex]\frac{1}{{a}^{18}{b}^{21}}[/latex]
• [latex]\frac{{r}^{12}}{{s}^{8}}[/latex]

Finding the Power of a Quotient

To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.

Let’s rewrite the original problem differently and look at the result.

It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.

The Power of a Quotient Rule of Exponents

For any real numbers[latex]\,a\,[/latex]and[latex]\,b\,[/latex]and any integer[latex]\,n,[/latex] the power of a quotient rule of exponents states that

Using the Power of a Quotient Rule

Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

• [latex]{\left(\frac{4}{{z}^{11}}\right)}^{3}[/latex]
• [latex]{\left(\frac{p}{{q}^{3}}\right)}^{6}[/latex]
• [latex]{\left(\frac{-1}{{t}^{2}}\right)}^{27}[/latex]
• [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}[/latex]
• [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}[/latex]
• [latex]{\left(\frac{4}{{z}^{11}}\right)}^{3}=\frac{{\left(4\right)}^{3}}{{\left({z}^{11}\right)}^{3}}=\frac{64}{{z}^{11\cdot 3}}=\frac{64}{{z}^{33}}[/latex]
• [latex]{\left(\frac{p}{{q}^{3}}\right)}^{6}=\frac{{\left(p\right)}^{6}}{{\left({q}^{3}\right)}^{6}}=\frac{{p}^{1\cdot 6}}{{q}^{3\cdot 6}}=\frac{{p}^{6}}{{q}^{18}}[/latex]
• [latex]{\left(\frac{-1}{{t}^{2}}\right)}^{27}=\frac{{\left(-1\right)}^{27}}{{\left({t}^{2}\right)}^{27}}=\frac{-1}{{t}^{2\cdot 27}}=\frac{-1}{{t}^{54}}=-\frac{1}{{t}^{54}}[/latex]
• [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}={\left(\frac{{j}^{3}}{{k}^{2}}\right)}^{4}=\frac{{\left({j}^{3}\right)}^{4}}{{\left({k}^{2}\right)}^{4}}=\frac{{j}^{3\cdot 4}}{{k}^{2\cdot 4}}=\frac{{j}^{12}}{{k}^{8}}[/latex]
• [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}={\left(\frac{1}{{m}^{2}{n}^{2}}\right)}^{3}=\frac{{\left(1\right)}^{3}}{{\left({m}^{2}{n}^{2}\right)}^{3}}=\frac{1}{{\left({m}^{2}\right)}^{3}{\left({n}^{2}\right)}^{3}}=\frac{1}{{m}^{2\cdot 3}\cdot {n}^{2\cdot 3}}=\frac{1}{{m}^{6}{n}^{6}}[/latex]
• [latex]{\left(\frac{{b}^{5}}{c}\right)}^{3}[/latex]
• [latex]{\left(\frac{5}{{u}^{8}}\right)}^{4}[/latex]
• [latex]{\left(\frac{-1}{{w}^{3}}\right)}^{35}[/latex]
• [latex]{\left({p}^{-4}{q}^{3}\right)}^{8}[/latex]
• [latex]{\left({c}^{-5}{d}^{-3}\right)}^{4}[/latex]
• [latex]\frac{{b}^{15}}{{c}^{3}}[/latex]
• [latex]\frac{625}{{u}^{32}}[/latex]
• [latex]\frac{-1}{{w}^{105}}[/latex]
• [latex]\frac{{q}^{24}}{{p}^{32}}[/latex]
• [latex]\frac{1}{{c}^{20}{d}^{12}}[/latex]

Simplifying Exponential Expressions

Recall that to simplify an expression means to rewrite it by combining terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.

Simplify each expression and write the answer with positive exponents only.

• [latex]{\left(6{m}^{2}{n}^{-1}\right)}^{3}[/latex]
• [latex]{17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}[/latex]
• [latex]{\left(\frac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}[/latex]
• [latex]\left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)[/latex]
• [latex]{\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}[/latex]
• [latex]\frac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}[/latex]
• [latex]\begin{array}{cccc}\hfill {\left(6{m}^{2}{n}^{-1}\right)}^{3}& =& {\left(6\right)}^{3}{\left({m}^{2}\right)}^{3}{\left({n}^{-1}\right)}^{3}\hfill & \phantom{\rule{9em}{0ex}}\text{The power of a product rule}\hfill \\ & =& {6}^{3}{m}^{2\cdot 3}{n}^{-1\cdot 3}\hfill & \phantom{\rule{9em}{0ex}}\text{The power rule}\hfill \\ & =& \text{ }216{m}^{6}{n}^{-3}\hfill & \phantom{\rule{9em}{0ex}}\text{Simplify}.\hfill \\ & =& \frac{216{m}^{6}}{{n}^{3}}\hfill & \phantom{\rule{9em}{0ex}}\text{The negative exponent rule}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill {17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}& =& {17}^{5-4-3}\hfill & \phantom{\rule{9em}{0ex}}\text{The product rule}\hfill \\ & =& {17}^{-2}\hfill & \phantom{\rule{9em}{0ex}}\text{Simplify}.\hfill \\ & =& \frac{1}{{17}^{2}}\text{ or }\frac{1}{289}\hfill & \phantom{\rule{9em}{0ex}}\text{The negative exponent rule}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill {\left(\frac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}& =& \frac{{\left({u}^{-1}v\right)}^{2}}{{\left({v}^{-1}\right)}^{2}}\hfill & \phantom{\rule{12em}{0ex}}\text{The power of a quotient rule}\hfill \\ & =& \frac{{u}^{-2}{v}^{2}}{{v}^{-2}}\hfill & \phantom{\rule{12em}{0ex}}\text{The power of a product rule}\hfill \\ & =& {u}^{-2}{v}^{2-\left(-2\right)}& \phantom{\rule{12em}{0ex}}\text{The quotient rule}\hfill \\ & =& {u}^{-2}{v}^{4}\hfill & \phantom{\rule{12em}{0ex}}\text{Simplify}.\hfill \\ & =& \frac{{v}^{4}}{{u}^{2}}\hfill & \phantom{\rule{12em}{0ex}}\text{The negative exponent rule}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill \left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)& =& -2\cdot 5\cdot {a}^{3}\cdot {a}^{-2}\cdot {b}^{-1}\cdot {b}^{2}\hfill & \phantom{\rule{3em}{0ex}}\text{Commutative and associative laws of multiplication}\hfill \\ & =& -10\cdot {a}^{3-2}\cdot {b}^{-1+2}\hfill & \phantom{\rule{3em}{0ex}}\text{The product rule}\hfill \\ & =& -10ab\hfill & \phantom{\rule{3em}{0ex}}\text{Simplify}.\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill {\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}& =& {\left({x}^{2}\sqrt{2}\right)}^{4-4}\hfill & \phantom{\rule{8em}{0ex}}\text{The product rule}\hfill \\ & =& \text{ }{\left({x}^{2}\sqrt{2}\right)}^{0}\hfill & \phantom{\rule{8em}{0ex}}\text{Simplify}.\hfill \\ & =& 1\hfill & \phantom{\rule{8em}{0ex}}\text{The zero exponent rule}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill \frac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}& =& \frac{{\left(3\right)}^{5}\cdot {\left({w}^{2}\right)}^{5}}{{\left(6\right)}^{2}\cdot {\left({w}^{-2}\right)}^{2}}\hfill & \phantom{\rule{13.5em}{0ex}}\text{The power of a product rule}\hfill \\ & =& \frac{{3}^{5}{w}^{2\cdot 5}}{{6}^{2}{w}^{-2\cdot 2}}\hfill & \phantom{\rule{13.5em}{0ex}}\text{The power rule}\hfill \\ & =& \frac{243{w}^{10}}{36{w}^{-4}}\hfill & \phantom{\rule{13.5em}{0ex}}\text{Simplify}.\hfill \\ & =& \frac{27{w}^{10-\left(-4\right)}}{4}\hfill & \phantom{\rule{13.5em}{0ex}}\text{The quotient rule and reduce fraction}\hfill \\ & =& \frac{27{w}^{14}}{4}\hfill & \phantom{\rule{13.5em}{0ex}}\text{Simplify}.\hfill \end{array}[/latex]
• [latex]{\left(2u{v}^{-2}\right)}^{-3}[/latex]
• [latex]{x}^{8}\cdot {x}^{-12}\cdot x[/latex]
• [latex]{\left(\frac{{e}^{2}{f}^{-3}}{{f}^{-1}}\right)}^{2}[/latex]
• [latex]\left(9{r}^{-5}{s}^{3}\right)\left(3{r}^{6}{s}^{-4}\right)[/latex]
• [latex]{\left(\frac{4}{9}t{w}^{-2}\right)}^{-3}{\left(\frac{4}{9}t{w}^{-2}\right)}^{3}[/latex]
• [latex]\frac{{\left(2{h}^{2}k\right)}^{4}}{{\left(7{h}^{-1}{k}^{2}\right)}^{2}}[/latex]
• [latex]\frac{{v}^{6}}{8{u}^{3}}[/latex]
• [latex]\frac{1}{{x}^{3}}[/latex]
• [latex]\frac{{e}^{4}}{{f}^{4}}[/latex]
• [latex]\frac{27r}{s}[/latex]
• [latex]\frac{16{h}^{10}}{49}[/latex]

Using Scientific Notation

Recall at the beginning of the section that we found the number[latex]\,1.3\,×\,{10}^{13}\,[/latex]when describing bits of information in digital images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an electron, which is about 0.00000000000047 m. How can we effectively work, read, compare, and calculate with numbers such as these?

A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places n that you moved the decimal point. Multiply the decimal number by 10 raised to a power of n . If you moved the decimal left as in a very large number,[latex]\,n\,[/latex]is positive. If you moved the decimal right as in a smaller large number,[latex]\,n\,[/latex]is negative.

For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.

We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.

Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.

Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

• Scientific Notation

A number is written in scientific notation if it is written in the form[latex]\,a\,×\,{10}^{n},[/latex] where[latex]\,1\le |a|10\,[/latex]and[latex]\,n\,[/latex]is an integer.

Converting Standard Notation to Scientific Notation

Write each number in scientific notation.

• Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m
• Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m
• Number of stars in Andromeda Galaxy: 1,000,000,000,000
• Diameter of electron: 0.00000000000094 m
• Probability of being struck by lightning in any single year: 0.00000143
• [latex]\begin{array}{l}24,000,000,000,000,000,000,000\text{ m}\hfill \\ \underset{←22\text{ places}}{\underset{}{24,000,000,000,000,000,000,000\text{ m}}}\hfill \\ 2.4\,×\,{10}^{22}\text{ m}\hfill \end{array}[/latex]
• [latex]\begin{array}{l}1,300,000,000,000,000,000,000\text{ m}\hfill \\ \underset{←21\text{ places}}{\underset{}{1,300,000,000,000,000,000,000\text{ m}}}\hfill \\ 1.3\,×\,{10}^{21}\text{ m}\hfill \end{array}[/latex]
• [latex]\begin{array}{l}1,000,000,000,000\hfill \\ \underset{←12\text{ places}}{\underset{}{1,000,000,000,000}}\hfill \\ 1\,×\,{10}^{12}\hfill \end{array}[/latex]
• [latex]\begin{array}{l}0.00000000000094\text{ m}\hfill \\ \underset{\to 13\text{ places}}{\underset{}{0.00000000000094\text{ m}}}\hfill \\ 9.4\,×\,{10}^{-13}\text{ m}\hfill \end{array}[/latex]
• [latex]\begin{array}{l}0.00000143\hfill \\ \underset{\to 6\text{ places}}{\underset{}{0.00000143}}\hfill \\ 1.43\,×\,{10}^{-6}\hfill \end{array}[/latex]

Observe that if the given number is greater than 1, as in examples a–c, the exponent of 10 is positive, and if the number is less than 1, as in examples d–e, the exponent is negative.

• U.S. national debt per taxpayer (April 2014): $152,000
• World population (April 2014): 7,158,000,000
• World gross national income (April 2014): $85,500,000,000,000
• Time for light to travel 1 m: 0.00000000334 s
• Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715
• [latex]$1.52×{10}^{5}[/latex]
• [latex]7.158×{10}^{9}[/latex]
• [latex]$8.55×{10}^{13}[/latex]
• [latex]3.34×{10}^{-9}[/latex]
• [latex]7.15×{10}^{-8}[/latex]

Converting from Scientific to Standard Notation

To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal[latex]\,n\,[/latex]places to the right if[latex]\,n\,[/latex]is positive or[latex]\,n\,[/latex]places to the left if[latex]\,n\,[/latex]is negative, and add zeros as needed. Remember, if[latex]\,n\,[/latex]is positive, the value of the number is greater than 1, and if[latex]\,n\,[/latex]is negative, the value of the number is less than one.

Converting Scientific Notation to Standard Notation

Convert each number in scientific notation to standard notation.

• [latex]3.547\,×\,{10}^{14}[/latex]
• [latex]-2\,×\,{10}^{6}[/latex]
• [latex]7.91\,×\,{10}^{-7}[/latex]
• [latex]-8.05\,×\,{10}^{-12}[/latex]
• [latex]\begin{array}{l}3.547\,×\,{10}^{14}\hfill \\ \underset{\to 14\text{ places}}{\underset{}{3.54700000000000}}\hfill \\ 354,700,000,000,000\hfill \end{array}[/latex]
• [latex]\begin{array}{l}-2\,×\,{10}^{6}\hfill \\ \underset{\to 6\text{ places}}{\underset{}{-2.000000}}\hfill \\ -2,000,000\hfill \end{array}[/latex]
• [latex]\begin{array}{l}7.91\,×\,{10}^{-7}\hfill \\ \underset{\to 7\text{ places}}{\underset{}{0000007.91}}\hfill \\ 0.000000791\hfill \end{array}[/latex]
• [latex]\begin{array}{l}-8.05\,×\,{10}^{-12}\hfill \\ \underset{\to 12\text{ places}}{\underset{}{-000000000008.05}}\hfill \\ -0.00000000000805\hfill \end{array}[/latex]
• [latex]7.03\,×\,{10}^{5}[/latex]
• [latex]-8.16\,×\,{10}^{11}[/latex]
• [latex]-3.9\,×\,{10}^{-13}[/latex]
• [latex]8\,×\,{10}^{-6}[/latex]
• [latex]703,000[/latex]
• [latex]-816,000,000,000[/latex]
• [latex]-0.000\,000\,000\,000\,39[/latex]
• [latex]0.000008[/latex]

Using Scientific Notation in Applications

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around[latex]\,1.32\,×\,{10}^{21}\,[/latex]molecules of water, and 1 L of water holds about[latex]\,1.22\,×\,{10}^{4}\,[/latex]average drops. Therefore, there are approximately[latex]\,3\cdot \left(1.32\,×\,{10}^{21}\right)\cdot \left(1.22\,×\,{10}^{4}\right)\approx 4.83\,×\,{10}^{25}\,[/latex]atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!

When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consider the product[latex]\,\left(7\,×\,{10}^{4}\right)\cdot \left(5\,×\,{10}^{6}\right)=35\,×\,{10}^{10}.\,[/latex]The answer is not in proper scientific notation because 35 is greater than 10. Consider 35 as[latex]\,3.5\,×\,10.\,[/latex]That adds a ten to the exponent of the answer.

Perform the operations and write the answer in scientific notation.

• [latex]\left(8.14\,×\,{10}^{-7}\right)\left(6.5\,×\,{10}^{10}\right)[/latex]
• [latex]\left(4\,×\,{10}^{5}\right)÷\left(-1.52\,×\,{10}^{9}\right)[/latex]
• [latex]\left(2.7\,×\,{10}^{5}\right)\left(6.04\,×\,{10}^{13}\right)[/latex]
• [latex]\left(1.2\,×\,{10}^{8}\right)÷\left(9.6\,×\,{10}^{5}\right)[/latex]
• [latex]\left(3.33\,×\,{10}^{4}\right)\left(-1.05\,×\,{10}^{7}\right)\left(5.62\,×\,{10}^{5}\right)[/latex]
• [latex]\begin{array}{cccc}\hfill \left(8.14\,×\,{10}^{-7}\right)\left(6.5\,×\,{10}^{10}\right)& =& \left(8.14\,×\,6.5\right)\left({10}^{-7}\,×\,{10}^{10}\right)\hfill & \begin{array}{l}\phantom{\rule{3em}{0ex}}\text{Commutative and associative}\hfill \\ \phantom{\rule{3em}{0ex}}\text{properties of multiplication}\hfill \end{array}\hfill \\ & =& \left(52.91\right)\left({10}^{3}\right)\hfill & \phantom{\rule{3em}{0ex}}\text{Product rule of exponents}\hfill \\ & =& 5.291\,×\,{10}^{4}\hfill & \phantom{\rule{3em}{0ex}}\text{Scientific notation}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill \left(4\,×\,{10}^{5}\right)÷\left(-1.52\,×\,{10}^{9}\right)& =& \left(\frac{4}{-1.52}\right)\left(\frac{{10}^{5}}{{10}^{9}}\right)\hfill & \begin{array}{l}\phantom{\rule{8em}{0ex}}\text{Commutative and associative}\hfill \\ \phantom{\rule{8em}{0ex}}\text{properties of multiplication}\hfill \end{array}\hfill \\ & \approx & \left(-2.63\right)\left({10}^{-4}\right)\hfill & \phantom{\rule{8em}{0ex}}\text{Quotient rule of exponents}\hfill \\ & =& -2.63\,×\,{10}^{-4}\hfill & \phantom{\rule{8em}{0ex}}\text{Scientific notation}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill \left(2.7\,×\,{10}^{5}\right)\left(6.04\,×\,{10}^{13}\right)& =& \left(2.7\,×\,6.04\right)\left({10}^{5}\,×\,{10}^{13}\right)\hfill & \begin{array}{l}\phantom{\rule{4em}{0ex}}\text{Commutative and associative}\hfill \\ \phantom{\rule{4em}{0ex}}\text{properties of multiplication}\hfill \end{array}\hfill \\ & =& \left(16.308\right)\left({10}^{18}\right)\hfill & \phantom{\rule{4em}{0ex}}\text{Product rule of exponents}\hfill \\ & =& 1.6308\,×\,{10}^{19}\hfill & \phantom{\rule{4em}{0ex}}\text{Scientific notation}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill \left(1.2\,×\,{10}^{8}\right)÷\left(9.6\,×\,{10}^{5}\right)& =& \left(\frac{1.2}{9.6}\right)\left(\frac{{10}^{8}}{{10}^{5}}\right)\hfill & \begin{array}{l}\phantom{\rule{9em}{0ex}}\text{Commutative and associative}\hfill \\ \phantom{\rule{9em}{0ex}}\text{properties of multiplication}\hfill \end{array}\hfill \\ & =& \left(0.125\right)\left({10}^{3}\right)\hfill & \phantom{\rule{9em}{0ex}}\text{Quotient rule of exponents}\hfill \\ & =& 1.25\,×\,{10}^{2}\hfill & \phantom{\rule{9em}{0ex}}\text{Scientific notation}\hfill \end{array}[/latex]
• [latex]\begin{array}{ccc}\hfill \left(3.33\,×\,{10}^{4}\right)\left(-1.05\,×\,{10}^{7}\right)\left(5.62\,×\,{10}^{5}\right)& =& \left[3.33\,×\,\left(-1.05\right)\,×\,5.62\right]\left({10}^{4}\,×\,{10}^{7}\,×\,{10}^{5}\right)\hfill \\ & \approx & \left(-19.65\right)\left({10}^{16}\right)\hfill \\ & =& -1.965\,×\,{10}^{17}\hfill \end{array}[/latex]
• [latex]\left(-7.5\,×\,{10}^{8}\right)\left(1.13\,×\,{10}^{-2}\right)[/latex]
• [latex]\left(1.24\,×\,{10}^{11}\right)÷\left(1.55\,×\,{10}^{18}\right)[/latex]
• [latex]\left(3.72\,×\,{10}^{9}\right)\left(8\,×\,{10}^{3}\right)[/latex]
• [latex]\left(9.933\,×\,{10}^{23}\right)÷\left(-2.31\,×\,{10}^{17}\right)[/latex]
• [latex]\left(-6.04\,×\,{10}^{9}\right)\left(7.3\,×\,{10}^{2}\right)\left(-2.81\,×\,{10}^{2}\right)[/latex]
• [latex]-8.475\,×\,{10}^{6}[/latex]
• [latex]8\,×\,{10}^{-8}[/latex]
• [latex]2.976\,×\,{10}^{13}[/latex]
• [latex]-4.3\,×\,{10}^{6}[/latex]
• [latex]\approx 1.24\,×{10}^{15}[/latex]

Applying Scientific Notation to Solve Problems

In April 2014, the population of the United States was about 308,000,000 people. The national debt was about $17,547,000,000,000. Write each number in scientific notation, rounding figures to two decimal places, and find the amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.

The population was[latex]\,308,000,000=3.08\,×\,{10}^{8}.[/latex]

The national debt was[latex]\,\text{\$}17,547,000,000,000\approx \text{\$}1.75\,×\,{10}^{13}.[/latex]

To find the amount of debt per citizen, divide the national debt by the number of citizens.

The debt per citizen at the time was about[latex]\,\text{\$}5.7\,×\,{10}^{4},[/latex]or $57,000.

An average human body contains around 30,000,000,000,000 red blood cells. Each cell measures approximately 0.000008 m long. Write each number in scientific notation and find the total length if the cells were laid end-to-end. Write the answer in both scientific and standard notations.

Number of cells:[latex]\,3×{10}^{13};[/latex]length of a cell:[latex]\,8×{10}^{-6}\,[/latex]m; total length:[latex]\,2.4×{10}^{8}\,[/latex]m or[latex]\,240,000,000\,[/latex]m.

Access these online resources for additional instruction and practice with exponents and scientific notation.

• Exponential Notation
• Properties of Exponents
• Zero Exponent
• Simplify Exponent Expressions
• Quotient Rule for Exponents
• Converting to Decimal Notation

Key Equations

Rules of exponents, key concepts.

• Products of exponential expressions with the same base can be simplified by adding exponents.
• Quotients of exponential expressions with the same base can be simplified by subtracting exponents.
• Powers of exponential expressions with the same base can be simplified by multiplying exponents.
• An expression with exponent zero is defined as 1.
• An expression with a negative exponent is defined as a reciprocal.
• The power of a product of factors is the same as the product of the powers of the same factors.
• The power of a quotient of factors is the same as the quotient of the powers of the same factors.
• The rules for exponential expressions can be combined to simplify more complicated expressions.
• Scientific notation uses powers of 10 to simplify very large or very small numbers.
• Scientific notation may be used to simplify calculations with very large or very small numbers.

Section Exercises

• Is[latex]\,{2}^{3}\,[/latex]the same as[latex]\,{3}^{2}?\,[/latex]Explain.

No, the two expressions are not the same. An exponent tells how many times you multiply the base. So[latex]\,{2}^{3}\,[/latex]is the same as[latex]\,2×2×2,[/latex] which is 8.[latex]\,{3}^{2}\,[/latex]is the same as[latex]\,3×3,[/latex] which is 9.

• When can you add two exponents?
• What is the purpose of scientific notation?

It is a method of writing very small and very large numbers.

• Explain what a negative exponent does.

For the following exercises, simplify the given expression. Write answers with positive exponents.

• [latex]\,{9}^{2}\,[/latex]
• [latex]{15}^{-2}[/latex]
• [latex]{3}^{2}\,×\,{3}^{3}[/latex]
• [latex]{4}^{4}÷4[/latex]
• [latex]{\left({2}^{2}\right)}^{-2}[/latex]

[latex]\frac{1}{16}[/latex]

• [latex]{\left(5-8\right)}^{0}[/latex]
• [latex]{11}^{3}÷{11}^{4}[/latex]

[latex]\frac{1}{11}[/latex]

• [latex]{6}^{5}\,×\,{6}^{-7}[/latex]
• [latex]{\left({8}^{0}\right)}^{2}[/latex]
• [latex]{5}^{-2}÷{5}^{2}[/latex]

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.

• [latex]{4}^{2}\,×\,{4}^{3}÷{4}^{-4}[/latex]

[latex]{4}^{9}[/latex]

• [latex]\frac{{6}^{12}}{{6}^{9}}[/latex]
• [latex]{\left({12}^{3}\,×\,12\right)}^{10}[/latex]

[latex]{12}^{40}[/latex]

• [latex]{10}^{6}÷{\left({10}^{10}\right)}^{-2}[/latex]
• [latex]{7}^{-6}\,×\,{7}^{-3}[/latex]

[latex]\frac{1}{{7}^{9}}[/latex]

• [latex]{\left({3}^{3}÷{3}^{4}\right)}^{5}[/latex]

For the following exercises, express the decimal in scientific notation.

[latex]3.14\,×{10}^{-5}[/latex]

• 148,000,000

For the following exercises, convert each number in scientific notation to standard notation.

• [latex]1.6\,×\,{10}^{10}[/latex]

16,000,000,000

• [latex]9.8\,×\,{10}^{-9}[/latex]
• [latex]\frac{{a}^{3}{a}^{2}}{a}[/latex]

[latex]{a}^{4}[/latex]

• [latex]\frac{m{n}^{2}}{{m}^{-2}}[/latex]
• [latex]{\left({b}^{3}{c}^{4}\right)}^{2}[/latex]

[latex]{b}^{6}{c}^{8}[/latex]

• [latex]{\left(\frac{{x}^{-3}}{{y}^{2}}\right)}^{-5}[/latex]
• [latex]a{b}^{2}÷{d}^{-3}[/latex]

[latex]a{b}^{2}{d}^{3}[/latex]

• [latex]{\left({w}^{0}{x}^{5}\right)}^{-1}[/latex]
• [latex]\frac{{m}^{4}}{{n}^{0}}[/latex]

[latex]{m}^{4}[/latex]

• [latex]{y}^{-4}{\left({y}^{2}\right)}^{2}[/latex]
• [latex]\frac{{p}^{-4}{q}^{2}}{{p}^{2}{q}^{-3}}[/latex]

[latex]\frac{{q}^{5}}{{p}^{6}}[/latex]

• [latex]{\left(l\,×\,w\right)}^{2}[/latex]
• [latex]{\left({y}^{7}\right)}^{3}÷{x}^{14}[/latex]

[latex]\frac{{y}^{21}}{{x}^{14}}[/latex]

• [latex]{\left(\frac{a}{{2}^{3}}\right)}^{2}[/latex]
• [latex]{5}^{2}m÷{5}^{0}m[/latex]

[latex]25[/latex]

• [latex]\frac{{\left(16\sqrt{x}\right)}^{2}}{{y}^{-1}}[/latex]
• [latex]\frac{{2}^{3}}{{\left(3a\right)}^{-2}}[/latex]

[latex]72{a}^{2}[/latex]

• [latex]{\left(m{a}^{6}\right)}^{2}\frac{1}{{m}^{3}{a}^{2}}[/latex]
• [latex]{\left({b}^{-3}c\right)}^{3}[/latex]

[latex]\frac{{c}^{3}}{{b}^{9}}[/latex]

• [latex]{\left({x}^{2}{y}^{13}÷{y}^{0}\right)}^{2}[/latex]
• [latex]{\left(9{z}^{3}\right)}^{-2}y[/latex]

[latex]\frac{y}{81{z}^{6}}[/latex]

Real-World Applications

• To reach escape velocity, a rocket must travel at the rate of[latex]\,2.2\,×\,{10}^{6}\,[/latex]ft/min. Rewrite the rate in standard notation.
• A dime is the thinnest coin in U.S. currency. A dime’s thickness measures[latex]\,1.35\,×\,{10}^{-3}\,[/latex]m. Rewrite the number in standard notation.
• The average distance between Earth and the Sun is 92,960,000 mi. Rewrite the distance using scientific notation.
• A terabyte is made of approximately 1,099,500,000,000 bytes. Rewrite in scientific notation.

[latex]1.0995×{10}^{12}[/latex]

• The Gross Domestic Product (GDP) for the United States in the first quarter of 2014 was[latex]\,\text{\$}1.71496\,×\,{10}^{13}.\,[/latex]Rewrite the GDP in standard notation.
• One picometer is approximately[latex]\,3.397\,×\,{10}^{-11}\,[/latex]in. Rewrite this length using standard notation.

0.00000000003397 in.

• The value of the services sector of the U.S. economy in the first quarter of 2012 was $10,633.6 billion. Rewrite this amount in scientific notation.

For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.

• [latex]{\left(\frac{{12}^{3}{m}^{33}}{{4}^{-3}}\right)}^{2}[/latex]

12,230,590,464[latex]\,{m}^{66}[/latex]

• [latex]{17}^{3}÷{15}^{2}{x}^{3}[/latex]
• [latex]{\left(\frac{{3}^{2}}{{a}^{3}}\right)}^{-2}{\left(\frac{{a}^{4}}{{2}^{2}}\right)}^{2}[/latex]

[latex]\frac{{a}^{14}}{1296}[/latex]

• [latex]{\left({6}^{2}-24\right)}^{2}÷{\left(\frac{x}{y}\right)}^{-5}[/latex]
• [latex]\frac{{m}^{2}{n}^{3}}{{a}^{2}{c}^{-3}}\cdot \frac{{a}^{-7}{n}^{-2}}{{m}^{2}{c}^{4}}[/latex]

[latex]\frac{n}{{a}^{9}c}[/latex]

• [latex]{\left(\frac{{x}^{6}{y}^{3}}{{x}^{3}{y}^{-3}}\cdot \frac{{y}^{-7}}{{x}^{-3}}\right)}^{10}[/latex]
• [latex]{\left(\frac{{\left(a{b}^{2}c\right)}^{-3}}{{b}^{-3}}\right)}^{2}[/latex]

[latex]\frac{1}{{a}^{6}{b}^{6}{c}^{6}}[/latex]

• Avogadro’s constant is used to calculate the number of particles in a mole. A mole is a basic unit in chemistry to measure the amount of a substance. The constant is[latex]\,6.0221413\,×\,{10}^{23}.\,[/latex]Write Avogadro’s constant in standard notation.
• Planck’s constant is an important unit of measure in quantum physics. It describes the relationship between energy and frequency. The constant is written as[latex]\,6.62606957\,×\,{10}^{-34}.\,[/latex]Write Planck’s constant in standard notation.

0.000000000000000000000000000000000662606957

Media Attributions

• 1.2 Scientific Notation I © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 1.2 Scientific Notation II © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license

College Algebra Copyright © 2024 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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8th grade (Illustrative Mathematics)

Course: 8th grade (illustrative mathematics)   >   unit 7.

• Unit test Exponents and scientific notation

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5.5: Integer Exponents and Scientific Notation

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Learning Objectives

By the end of this section, you will be able to:

• Use the definition of a negative exponent
• Simplify expressions with integer exponents
• Convert from decimal notation to scientific notation
• Convert scientific notation to decimal form
• Multiply and divide using scientific notation

Before you get started, take this readiness quiz.

• What is the place value of the 6 in the number 64891? If you missed this problem, review Example 1.2.1 .
• Name the decimal: 0.0012. If you missed this problem, review Example 1.8.1 .
• Subtract: 5−(−3). If you missed this problem, review Example 1.4.33 .

Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents introduced earlier in this chapter, has two forms depending on whether the exponent is larger in the numerator or the denominator.

QUOTIENT PROPERTY FOR EXPONENTS

If a is a real number, \(a\neq0\), and m and n are whole numbers, then

\[\dfrac{a^{m}}{a^{n}}=a^{m-n}, m>n \quad\]

\[\dfrac{a^{m}}{a^{n}}=\dfrac{1}{a^{n-m}}, n>m\]

What if we just subtract exponents regardless of which is larger?

Let’s consider \(\dfrac{x^{2}}{x^{5}}\).

We subtract the exponent in the denominator from the exponent in the numerator.

\[\begin{array}{c}{\dfrac{x^{2}}{x^{5}}} \\ {x^{2-5}} \\ {x^{-3}}\end{array}\]

We can also simplify \(\dfrac{x^{2}}{x^{5}}\) by dividing out common factors:

This implies that \(x^{-3}=\dfrac{1}{x^{3}}\) and it leads us to the definition of a negative exponent .

Definition: NEGATIVE EXPONENT

If n is an integer and \(a\neq 0\), then \(a^{-n}=\dfrac{1}{a^{n}}\)

The negative exponent tells us we can re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent.

Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.

For example, if after simplifying an expression we end up with the expression \(x^{-3}\), we will take one more step and write \(\dfrac{1}{x^{3}}\). The answer is considered to be in simplest form when it has only positive exponents.

Example \(\PageIndex{1}\)

• \(10^{-3}\)

\(\begin{array}{lll} 1. && 4^{-2} \\& {\text { Use the definition of a negative exponent, } a^{-n}=\dfrac{1}{a^{n}},} & {\dfrac{1}{4^{2}}} \\& {\text { Simplify. }} & \dfrac{1}{16} \end{array}\)

\(\begin{array}{lll} \\ 2. && 10^{-3} \\& {\text { Use the definition of a negative exponent, } a^{-n}=\dfrac{1}{a^{n}},} & \dfrac{1}{10^{3}} \\&  {\text { Simplify. }} & \dfrac{1}{1000}\end{array}\)

Try It \(\PageIndex{2}\)

• \(10^{-7}\)
• \(\dfrac{1}{8}\)
• \(\dfrac{1}{10^{7}}\)

Try It \(\PageIndex{3}\)

• \(10^{-4}\)
• \(\dfrac{1}{9}\)
• \(\dfrac{1}{10,000}\)

In Example \(\PageIndex{1}\) we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We’ll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

\(\begin{array}{ll}& \dfrac{1}{a^{-n}}\\ {\text { Use the definition of a negative exponent, } a^{-n}=\dfrac{1}{a^{n}} } & \dfrac{1}{\dfrac{1}{a^{n}}} \\ {\text { Simplify the complex fraction. }} & 1 \cdot \dfrac{a^{n}}{1}\\ {\text { Multiply. }} & a^{n}\end{array}\)

This leads to the Property of Negative Exponents.

PROPERTY OF NEGATIVE EXPONENTS

If n is an integer and \(a\neq 0\), then \(\dfrac{1}{a^{-n}}=a^{n}\).

Example \(\PageIndex{4}\)

• \(\dfrac{1}{y^{-4}}\)
• \(\dfrac{1}{3^{-2}}\)

\(\begin{array} { lll } 1. && \dfrac{1}{y^{-4}}\\& \text { Use the property of a negative exponent, } \dfrac{1}{a^{-n}}=a^{n} . & y^{4}\end{array}\)

\(\begin{array} { lll } \\ 2. && \dfrac{1}{3^{-2}}\\& \text {Use the property of a negative exponent, } \dfrac{1}{a^{-n}}=a^{n} . & 3^{2} \\& \text{Simplify.}& 9\end{array}\)

Try It \(\PageIndex{5}\)

• \(\dfrac{1}{p^{-8}}\)
• \(\dfrac{1}{4^{-3}}\)

Try It \(\PageIndex{6}\)

• \(\dfrac{1}{q^{-7}}\)
• \(\dfrac{1}{2^{-4}}\)

Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.

\(\begin{array}{ll}& \left(\dfrac{3}{4}\right)^{-2}\\ {\text { Use the definition of a negative exponent, } a^{-n}=\dfrac{1}{a^{n}} } & \dfrac{1}{\left(\dfrac{3}{4}\right)^{2}} \\ {\text { Simplify the denominator. }} & \dfrac{1}{\dfrac{9}{16}}\\ {\text { Simplify the complex fraction.}} &\dfrac{16}{9}\\ \text { But we know that } \dfrac{16}{9} \text { is }\left(\dfrac{4}{3}\right)^{2} & \\ \text { This tells us that: } & \left(\dfrac{3}{4}\right)^{-2}=\left(\dfrac{4}{3}\right)^{2}\end{array}\)

To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.

This leads us to the Quotient to a Negative Power Property .

QUOTIENT TO A NEGATIVE EXPONENT PROPERTY

If \(a\) and \(b\) are real numbers, \(a \neq 0, b \neq 0,\) and \(n\) is an integer, then \(\left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}\)

Example \(\PageIndex{7}\)

• \(\left(\dfrac{5}{7}\right)^{-2}\)
• \( \left(-\dfrac{2 x}{y}\right)^{-3} \)

\(\begin{array}{lll}1. && \left(\dfrac{5}{7}\right)^{-2}\\ & \begin{array}{l}       {\text {Use the Quotient to a Negative Exponent Property, }  } \\       { \left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}  \text {.  Take the reciprocal of the }  } \\       {\text {fraction and change the sign of the exponent. }. }        \end{array} &\left(\dfrac{7}{5}\right)^{2}\\ & \text { Simplify. } & \dfrac{49}{25}\end{array}\)

\(\begin{array}{lll}\\2. && \left(-\dfrac{2 x}{y}\right)^{-3}\\ & \begin{array}{l}       {\text {Use the Quotient to a Negative Exponent Property, }  } \\       { \left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}  \text {.  Take the reciprocal of the }  } \\       {\text {fraction and change the sign of the exponent. } }        \end{array} &\left(-\dfrac{y}{2 x}\right)^{3}\\ & \text { Simplify. } & -\dfrac{y^{3}}{8 x^{3}}\end{array}\)

Try It \(\PageIndex{8}\)

• \(\left(\dfrac{2}{3}\right)^{-4}\)
• \(\left(-\dfrac{6 m}{n}\right)^{-2}\)
• \(\dfrac{81}{16} \)
• \(\dfrac{n^{2}}{36 m^{2}}\)

Try It \(\PageIndex{9}\)

• \(\left(\dfrac{3}{5}\right)^{-3}\)
• \(\left(-\dfrac{a}{2 b}\right)^{-4}\)
• \(\dfrac{125}{27}\)
• \(\dfrac{16 b^{4}}{a^{4}}\)

When simplifying an expression with exponents, we must be careful to correctly identify the base.

Example \(\PageIndex{10}\)

• \((-3)^{-2}\)
• \(-3^{-2}\)
• \(\left(-\dfrac{1}{3}\right)^{-2}\)
• \(-\left(\dfrac{1}{3}\right)^{-2}\)

1. Here the exponent applies to the base −3.

\(\begin{array}{ll} & (-3)^{-2}\\ \begin{array}{l}       {\text { Take the reciprocal of the base }     }\\       {\text { and change the sign of the exponent}. }       \end{array} & \dfrac{1}{(-3)^{2}} \\ {\text { Simplify. }} & \dfrac{1}{9}\end{array}\)

2. The expression \(-3^{-2}\) means “find the opposite of \(3^{-2}\)”. Here the exponent applies to the base 3.

\(\begin{array}{ll} &-3^{-2}\\ \text { Rewrite as a product with }-1&-1 \cdot 3^{-2}\\ \begin{array}{l}       {\text { Take the reciprocal of the base }     }\\       {\text { and change the sign of the exponent}. }       \end{array} & -1 \cdot \dfrac{1}{3^{2}}\\ {\text { Simplify. }} & -\dfrac{1}{9}\end{array}\)

3. Here the exponent applies to the base\(\left(-\frac{1}{3}\right)\).

\(\begin{array}{ll} &\left(-\dfrac{1}{3}\right)^{-2}\\ \begin{array}{l}       {\text { Take the reciprocal of the base }     }\\       {\text { and change the sign of the exponent}. }       \end{array} & \left(-\dfrac{3}{1}\right)^{2}\\ {\text { Simplify. }} & 9\end{array}\)

4. The expression \(-\left(\frac{1}{3}\right)^{-2}\) means “find the opposite of \(\left(\frac{1}{3}\right)^{-2}\)”. Here the exponent applies to the base \(\left(\frac{1}{3}\right)\).

\(\begin{array}{ll} &-\left(\dfrac{1}{3}\right)^{-2}\\ \text { Rewrite as a product with }-1&-1 \cdot\left(\dfrac{1}{3}\right)^{-2}\\ \begin{array}{l}       {\text { Take the reciprocal of the base }     }\\       {\text { and change the sign of the exponent}. }       \end{array} & -1 \cdot\left(\dfrac{3}{1}\right)^{2}\\ {\text { Simplify. }} & -9 \end{array}\)

Try It \(\PageIndex{11}\)

• \((-5)^{-2}\)
• \(-5^{-2}\)
• \(\left(-\dfrac{1}{5}\right)^{-2}\)
• \(-\left(\dfrac{1}{5}\right)^{-2}\)
• \(\dfrac{1}{25}\)
• \(-\dfrac{1}{25}\)
• \(−25\)

Try It \(\PageIndex{12}\)

• \((-7)^{-2}\)
• \(-7^{-2}\)
• \(\left(-\dfrac{1}{7}\right)^{-2}\)
• \(-\left(\dfrac{1}{7}\right)^{-2}\)
• \(\dfrac{1}{49}\)
• \(-\dfrac{1}{49}\)
• \(−49\)

We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different.

Example \(\PageIndex{13}\)

• 4\(\cdot 2^{-1}\)
• \((4 \cdot 2)^{-1}\)

\(\begin{array}{lll} 1. &\text { Do exponents before multiplication. }&4 \cdot 2^{-1}\\& \text { Use } a^{-n}=\dfrac{1}{a^{n}}&4 \cdot \dfrac{1}{2^{1}}\\& {\text { Simplify. }} & 2 \end{array}\)

\(\begin{array}{lll} \\2. &&(4 \cdot 2)^{-1}\\& \text { Simplify inside the parentheses first. }&(8)^{-1}\\& \text { Use } a^{-n}=\dfrac{1}{a^{n}} & \dfrac{1}{8^{1}}\\&{\text { Simplify. }} & \dfrac{1}{8} \end{array}\)

Try It \(\PageIndex{14}\)

• 6\(\cdot 3^{-1}\)
• \((6 \cdot 3)^{-1}\)
• \(\dfrac{1}{18}\)

Try It \(\PageIndex{15}\)

• 8\(\cdot 2^{-2}\)
• \((8 \cdot 2)^{-2}\)
• \(\dfrac{1}{256}\)

When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. We will assume all variables are non-zero.

Example \(\PageIndex{16}\)

• \(\left(u^{4}\right)^{-3}\)

\(\begin{array}{lll}1. & &x^{-6}\\& \text { Use the definition of a negative exponent, } a^{-n}=\dfrac{1}{a^{n}}&\dfrac{1}{x^{6}}\end{array}\)

\(\begin{array}{lll} \\ 2. &\left(u^{4}\right)^{-3}\\& \text { Use the definition of a negative exponent, } a^{-n}=\dfrac{1}{a^{n}}&\dfrac{1}{\left(u^{4}\right)^{3}} \\& \text{ Simplify.} & \dfrac{1}{u^{12}}\end{array}\)

Try It \(\PageIndex{17}\)

• \(\left(z^{3}\right)^{-5}\)
• \(\dfrac{1}{y^{7}}\)
• \(\dfrac{1}{z^{15}}\)

Try It \(\PageIndex{18}\)

• \(\left(q^{4}\right)^{-6}\)
• \(\dfrac{1}{p^{9}}\)
• \(\dfrac{1}{q^{24}}\)

When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the Order of Operations, we simplify expressions in parentheses before applying exponents. We’ll see how this works in the next example.

Example \(\PageIndex{19}\)

• \( 5 y^{-1}\)
• \((5 y)^{-1}\)
• \((-5 y)^{-1}\)

1. Notice the exponent applies to just the base \(y\).

\(\begin{array}{ll} & 5 y^{-1}\\ \begin{array}{l}       {\text { Take the reciprocal of the base } y,    } \\       {\text { and change the sign of the exponent}. }        \end{array} &5 \cdot \dfrac{1}{y^{1}} \\ \text { Simplify. } & \dfrac{5}{y}\end{array}\)

2. Here the parentheses make the exponent apply to the base \(5y\)

\(\begin{array}{ll} &(5 y)^{-1}\\ \begin{array}{l}       {\text { Take the reciprocal of the base } 5y,    }\\       {\text { and change the sign of the exponent}. }        \end{array}  &\dfrac{1}{(5 y)^{1}}\\ \text { Simplify. } &\dfrac{1}{5 y} \end{array}\)

3. The base here is \( -5 y \)

\(\begin{array}{ll}   &(-5 y)^{-1}\\ \begin{array}{l}       {\text { Take the reciprocal of the base }-5 y    }\\       {\text { and change the sign of the exponent}. }        \end{array}  & \dfrac{1}{(-5 y)^{1}} \\ \text { Simplify. } & \dfrac{1}{-5 y} \\ \text { Use } \dfrac{a}{-b}=-\dfrac{a}{b} & -\dfrac{1}{5 y} \end{array}\)

Try It \(\PageIndex{20}\)

• \(8 p^{-1}\)
• \((8 p)^{-1}\)
• \((-8 p)^{-1}\)
• \(\dfrac{8}{p}\)
• \(\dfrac{1}{8 p}\)
• \(-\dfrac{1}{8 p}\)

Try It \(\PageIndex{21}\)

• \(11 q^{-1}\)
• \((11 q)^{-1} \)
• \((-11 q)^{-1}\)
• \(\dfrac{11}{q}\)
• \(\dfrac{1}{11 q} \)
• \(-\dfrac{1}{11 q}\)

With negative exponents, the Quotient Rule needs only one form \(\dfrac{a^{m}}{a^{n}}=a^{m-n},\) for \(a \neq 0\). When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative.

Simplify Expressions with Integer Exponents

All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

SUMMARY OF EXPONENT PROPERTIES

If \(a\) and \(b\) are real numbers, and \(m\) and \(n\) are integers, then

\(\begin{array}{lrll}{\textbf { Product Property }}& a^{m} \cdot a^{n} &=&a^{m+n} \\ {\textbf { Power Property }} &\left(a^{m}\right)^{n} &=&a^{m \cdot n} \\ {\textbf { Product to a Power }} &(a b)^{m} &=&a^{m} b^{m} \\ {\textbf { Quotient Property }} & \dfrac{a^{m}}{a^{n}} &=&a^{m-n}, \quad a \neq 0 \\ {\textbf { Zero Exponent Property }}& \quad a^{0} &= & 1, \quad a \neq 0 \\ {\textbf { Quotient to a Power Property }} & \left(\dfrac{a}{b}\right)^{m} &=&\dfrac{a^{m}}{b^{m}},\quad  b \neq 0 \\ {\textbf { Properties of Negative Exponents }} & a^{-n} &=&\dfrac{1}{a^{n}} \text { and } \dfrac{1}{a^{-n}}=a^{n}\\ {\textbf { Quotient to a Negative Exponents }}& \left(\dfrac{a}{b}\right)^{-n} &=&\left(\dfrac{b}{a}\right)^{n} \\\end{array}\)

Example \(\PageIndex{22}\)

• \(x^{-4} \cdot x^{6}\)
• \(y^{-6} \cdot y^{4}\)
• \(z^{-5} \cdot z^{-3}\)

\(\begin{array}{lll} 1. && x^{-4} \cdot x^{6} \\ & \text { Use the Product Property, } a^{m} \cdot a^{n}=a^{m+n} & x^{-4+6} \\ & \text { Simplify. } & x^{2} \end{array}\)

\(\begin{array}{lll} \\2. && y^{-6} \cdot y^{4} \\ & \text { Notice the same bases, so add the exponents. }& y^{-6+4}\\& \text { Simplify. } & y^{-2} \\ & \text { Use the definition of a negative exponent, } a^{-n}=\dfrac{1}{a^{n}} & \dfrac{1}{y^{2}} \end{array}\)

\(\begin{array}{lll} \\3. && z^{-5} \cdot z^{-3} \\ & \text { Add the exponents, since the bases are the same. }& z^{-5-3}\\ & \text { Simplify. } & z^{-8}\\ &  \begin{array}{l}       {\text {Use the definition of a negative exponent }     }\\       {\text { to take the reciprocal of the base }         }\\       {\text { and change the sign of the exponent}. }        \end{array}  & \dfrac{1}{z^{8}} \\ \end{array}\)

Try It \(\PageIndex{23}\)

• \(x^{-3} \cdot x^{7}\)
• \(y^{-7} \cdot y^{2}\)
• \(z^{-4} \cdot z^{-5}\)
• \(\dfrac{1}{y^{5}}\)
• \(\dfrac{1}{z^{9}}\)

Try It \(\PageIndex{24}\)

• \(a^{-1} \cdot a^{6}\)
• \(b^{-8} \cdot b^{4}\)
• \(c^{-8} \cdot c^{-7}\)
• \(\dfrac{1}{b^{4}}\)
• \(\dfrac{1}{c^{15}}\)

In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property.

Example \(\PageIndex{25}\)

Simplify: \(\left(m^{4} n^{-3}\right)\left(m^{-5} n^{-2}\right)\)

\(\begin{array}{ll}& \left(m^{4} n^{-3}\right)\left(m^{-5} n^{-2}\right) \\ \text { Use the Commutative Property to get like bases together. }& m^{4} m^{-5} \cdot n^{-2} n^{-3}\\ \text { Add the exponents for each base. }&m^{-1} \cdot n^{-5}\\ \text { Take reciprocals and change the signs of the exponents. }& \dfrac{1}{m^{1}} \cdot \dfrac{1}{n^{5}} \\ \text { Simplify. } & \dfrac{1}{m n^{5}}\end{array}\)

Try It \(\PageIndex{26}\)

Simplify: \(\left(p^{6} q^{-2}\right)\left(p^{-9} q^{-1}\right)\)

\(\dfrac{1}{p^3 q^3}\)

Try It \(\PageIndex{27}\)

Simplify:\(\left(r^{5} s^{-3}\right)\left(r^{-7} s^{-5}\right)\)

\(\dfrac{1}{r^2 s^8}\)

If the monomials have numerical coefficients, we multiply the coefficients, just like we did earlier.

Example \(\PageIndex{28}\)

Simplify: \(\left(2 x^{-6} y^{8}\right)\left(-5 x^{5} y^{-3}\right)\)

\(\begin{array}{ll}& \left(2 x^{-6} y^{8}\right)\left(-5 x^{5} y^{-3}\right) \\ \text { Rewrite with the like bases together. }& 2(-5) \cdot\left(x^{-6} x^{5}\right) \cdot\left(y^{8} y^{-3}\right)\\ \text { Multiply the coefficients and add the exponents of each variable. }&-10 \cdot x^{-1} \cdot y^{5}\\ \text { Use the definition of a negative exponent, } a^{-n}=\dfrac{1}{a^{n}}&-10 \cdot \dfrac{1}{x^{1}} \cdot y^{5} \\ \text { Simplify. } & \dfrac{-10 y^{5}}{x}\end{array}\)

Try It \(\PageIndex{29}\)

Simplify: \(\left(3 u^{-5} v^{7}\right)\left(-4 u^{4} v^{-2}\right)\)

\(-\dfrac{12v^5}{u}\)

Try It \(\PageIndex{30}\)

Simplify: \(\left(-6 c^{-6} d^{4}\right)\left(-5 c^{-2} d^{-1}\right)\)

\(\dfrac{30d^3}{c^8}\)

In the next two examples, we’ll use the Power Property and the Product to a Power Property.

Example \(\PageIndex{31}\)

Simplify: \(\left(6 k^{3}\right)^{-2}\)

\(\begin{array}{ll}&\left(6 k^{3}\right)^{-2}\\ \text { Use the Product to a Power Property, }(a b)^{m}=a^{n} b^{m}&(6)^{-2}\left(k^{3}\right)^{-2}\\ \text { Use the Power Property, }\left(a^{m}\right)^{n}=a^{m \cdot n}&6^{-2} k^{-6}\\ \text { Use the definition of a negative exponent, } a^{-n}=\dfrac{1}{a^{n}}&\dfrac{1}{6^{2}} \cdot \dfrac{1}{k^{6}} \\ \text { Simplify. } & \dfrac{1}{36 k^{6}}\end{array}\)

Try It \(\PageIndex{32}\)

Simplify: \(\left(-4 x^{4}\right)^{-2}\)

\(\dfrac{1}{16x^8}\)

Try It \(\PageIndex{33}\)

Simplify: \(\left(2 b^{3}\right)^{-4}\)

\(\dfrac{1}{16b^{12}}\)

Example \(\PageIndex{34}\)

Simplify: \(\left(5 x^{-3}\right)^{2}\)

\(\begin{array}{ll}&\left(5 x^{-3}\right)^{2}\\ \text { Use the Product to a Power Property, }(a b)^{m}=a^{n} b^{m}&5^{2}\left(x^{-3}\right)^{2}\\ \begin{array}{l}{\text { Simplify } 5^{2} \text { and multiply the exponents of } x \text { using the Power }} \\ {\text { Property, }\left(a^{m}\right)^{n}=a^{m \cdot n} .}\end{array}&25 \cdot x^{-6}\\ \begin{array}{l}{\text { Rewrite } x^{-6} \text { by using the Definition of a Negative Exponent, }} \\ {\space a^{-n}=\dfrac{1}{a^{n}}}\end{array}&25 \cdot \dfrac{1}{x^{6}}\\ \text { Simplify. } & \dfrac{25}{x^{6}}\end{array}\)

Try It \(\PageIndex{35}\)

Simplify: \(\left(8 a^{-4}\right)^{2}\)

\(\dfrac{64}{a^8}\)

Try It \(\PageIndex{36}\)

Simplify: \(\left(2 c^{-4}\right)^{3}\)

\(\dfrac{8}{c^{12}}\)

To simplify a fraction, we use the Quotient Property and subtract the exponents.

Example \(\PageIndex{37}\)

Simplify: \(\dfrac{r^{5}}{r^{-4}}\)

\(\begin{array}{l} & \dfrac{r^{5}}{r^{-4}}\\ {\text { Use the Quotient Property, } \dfrac{a^{n}}{a^{n}}=a^{m-n}} & r^{5-(-4)}\\ {\text { Simplify. }} & r^{9}\end{array}\)

Try It \(\PageIndex{38}\)

Simplify: \(\dfrac{x^{8}}{x^{-3}}\)

Try It \(\PageIndex{39}\)

Simplify: \(\dfrac{y^{8}}{y^{-6}}\)

Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and 0.004. We know that 4,000 means \(4 \times 1,000\) and 0.004 means \(4 \times \dfrac{1}{1,000}\).

If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:

\[\begin{array}{ll}{4,000} & {0.004} \\ {4 \times 1,000} & {4 \times \dfrac{1}{1,000}} \\ {4 \times 10^{3}} & {4 \times \dfrac{1}{10^{3}}} \\ & {4 \times 10^{-3}}\end{array}\]

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

SCIENTIFIC NOTATION

A number is expressed in scientific notation when it is of the form

\[a \times 10^{n} \text { where } 1 \leq a<10 \text { and } n \text { is an integer }\]

It is customary in scientific notation to use as the \(\times\) multiplication sign, even though we avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.

\(\begin{array}{ll}{\text { The power of } 10 \text { is positive when the number is larger than } 1 :} & {4,000=4 \times 10^{3}} \\ {\text { The power of } 10 \text { is negative when the number is between } 0 \text { and } 1 :} & {0.004=4 \times 10^{-3}} \end{array}\)

Example \(\PageIndex{40}\): HOW TO CONVERT FROM DECIMAL NOTATION TO SCIENTIFIC NOTATION

Write in scientific notation: 37000.

Try It \(\PageIndex{41}\)

Write in scientific notation: 96000.

\(9.6 \times 10^{4}\)

Try It \(\PageIndex{42}\)

Write in scientific notation: 48300.

\(4.83 \times 10^{4}\)

HOW TO: Convert from decimal notation to scientific notation

• Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
• Step 2. Count the number of decimal places, n , that the decimal point was moved.
• greater than 1, the power of 10 will be 10 n .
• between 0 and 1, the power of 10 will be 10 −n .
• Step 4. Check.

Example \(\PageIndex{43}\)

Write in scientific notation: 0.0052.

The original number, 0.0052, is between 0 and 1 so we will have a negative power of 10.

Try It \(\PageIndex{44}\)

Write in scientific notation: 0.0078

\(7.8 \times 10^{-3}\)

Try It \(\PageIndex{45}\)

Write in scientific notation: 0.0129

\(1.29 \times 10^{-2}\)

Convert Scientific Notation to Decimal Form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

\[\begin{array}{cc}{9.12 \times 10^{4}} & {9.12 \times 10^{-4}} \\ {9.12 \times 10,000} & {9.12 \times 0.0001} \\ {91,200} & {0.000912}\end{array}\]

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

\[9.12 \times 10^{4}=91,200 \quad 9.12 \times 10^{-4}=0.000912\]

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

Example \(\PageIndex{46}\)

Convert to decimal form: \(6.2 \times 10^{3}\)

Try It \(\PageIndex{47}\)

Convert to decimal form: \(1.3 \times 10^{3}\)

Try It \(\PageIndex{48}\)

Convert to decimal form: \(9.25 \times 10^{4}\)

The steps are summarized below.

Convert scientific notation to decimal form.

To convert scientific notation to decimal form:

• Step 1. Determine the exponent, \(n\), on the factor \(10\).
• If the exponent is positive, move the decimal point \(n\) places to the right.
• If the exponent is negative, move the decimal point \(|n|\) places to the left.
• Step 3. Check.

Example \(\PageIndex{49}\)

Convert to decimal form: \(8.9\times 10^{-2}\)

Try It \(\PageIndex{50}\)

Convert to decimal form: \(1.2 \times 10^{-4}\)

\(0.00012\)

Try It \(\PageIndex{51}\)

Convert to decimal form: \(7.5 \times 10^{-2}\)

Multiply and Divide Using Scientific Notation

Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

Example \(\PageIndex{52}\)

Multiply. Write answers in decimal form:\(\left(4 \times 10^{5}\right)\left(2 \times 10^{-7}\right)\)

\(\begin{array}{ll} & \left(4 \times 10^{5}\right)\left(2 \times 10^{-7}\right)\\\text { Use the Commutative Property to rearrange the factors. }& 4 \cdot 2 \cdot 10^{5} \cdot 10^{-7} \\ \text{ Multiply.} & 8 \times 10^{-2} \\ \text { Change to decimal form by moving the decimal two places left. } & 0.08\end{array}\)

Try It \(\PageIndex{53}\)

Multiply \((3\times 10^{6})(2\times 10^{-8})\). Write answers in decimal form.

Try It \(\PageIndex{54}\)

Multiply \(\left(3 \times 10^{-2}\right)\left(3 \times 10^{-1}\right)\). Write answers in decimal form.

Example \(\PageIndex{55}\)

Divide. Write answers in decimal form: \(\dfrac{9 \times 10^{3}}{3 \times 10^{-2}}\)

\(\begin{array}{ll} & \dfrac{9 \times 10^{3}}{3 \times 10^{-2}}\\\text { Separate the factors, rewriting as the product of two fractions. }& \dfrac{9}{3} \times \dfrac{10^{3}}{10^{-2}}\\ \text{ Divide.} & 3 \times 10^{5} \\ \text { Change to decimal form by moving the decimal five places right. } & 300000\end{array}\)

Try It \(\PageIndex{56}\)

Divide \(\dfrac{8 \times 10^{4}}{2 \times 10^{-1}} .\) Write answers in decimal form.

\(400,000\)

Try It \(\PageIndex{57}\)

Divide \(\dfrac{8 \times 10^{2}}{4 \times 10^{-2}} .\) Write answers in decimal form.

MEDIA ACCESS ADDITIONAL ONLINE RESOURCES

Access these online resources for additional instruction and practice with integer exponents and scientific notation:

• Negative Exponents
• Scientific Notation
• Scientific Notation 2

Key Concepts

• If \(n\) is a positive integer and \(a \ne 0\), then \(\dfrac{1}{a^{−n}}=a^n\)
• If \(a\) and \(b\) are real numbers, \(b \ne 0\) and \(n\) is an integer , then \(\left(\dfrac{a}{b}\right)^{−n}=\left(\dfrac{b}{a}\right)^n\)
• Determine the exponent, \(n\) on the factor \(10\).
• Move the decimal point so that the first factor is greater than or equal to \(1\) but less than \(10\).
• Count the number of decimal places, \(n\) that the decimal point was moved.
• greater than \(1\), the power of \(10\) will be \(10^n\)
• between \(0\) and \(1\), the power of \(10\) will be \(10^{−n}\)

Curriculum  /  Math  /  8th Grade  /  Unit 1: Exponents and Scientific Notation  /  Lesson 9

Exponents and Scientific Notation

Lesson 9 of 15

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Simplify and write equivalent exponential expressions using all exponent rules.

Common Core Standards

Core standards.

The core standards covered in this lesson

Expressions and Equations

8.EE.A.1 — Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3^-5 = 3^-3 = 1/3³ = 1/27.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Apply any or all exponent rules to write equivalent exponential expressions that are simplified or use fewer bases or factors. 
• Justify each step in order to know that each expression is equivalent.
• Reason with positive and negative bases and exponents.

Suggestions for teachers to help them teach this lesson

The goal of this lesson is for students to continue practicing with writing equivalent exponent expressions, now using all the rules and strategies they have learned. This can also be used as a flex day, depending on the needs of your particular students. The problem set guidance includes options for independent work as well as whole class or small group activities.

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Find $$n$$ so that the number sentence below is true:

$${2^3\cdot4^3 = 2^3\cdot2^n=2^9}$$

Then use your response above to explain why  $${2^3\cdot4^3=2^9}$$ .

Guiding Questions

Grade 8 Mathematics > Module 1 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds . © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US  license. Accessed Dec. 2, 2016, 5:17 p.m..

Write a simplified, equivalent expression for the one shown below:

$${4x^2(2y^2)^{-1}\over{2x^{-2}y^0}}$$

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

If $$x$$ is a positive integer greater than 1, then which of the following will be positive? Select all that apply.

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Use any other problems not used from previous lessons.

• RDA Performance Task Bank Grade 8 Mathematics Sample SR Item
• Illustrative Mathematics Extending the Definition of Exponents, Variation 1 — Challenge
• Don’t Panic, the Answer is 42 Exponent Rules Unit — Good options to use include the Exponent Rules Proof Worksheets, the Exponent Rules Karuta game, and the Exponent Quizzes.
• EngageNY Mathematics Grade 8 Mathematics > Module 1 > Topic B > Lesson 8 — Applying Properties of Exponents to Generate Equivalent Expressions - Round 1 & 2
• Yummy Math Negative Exponents? Ugh — This is set in a real-world context and is an interesting application for students to work with exponents in context.)
• MARS Formative Assessment Lesson for Grade 8 Applying Properties of Exponents
• Math Education Page Math Curriculum Materials for Middle School — Negative Exponents, 8th Grade (Includes problems that require students to explain patterns and rules, not just rote practice.)

Topic A: Review of Exponents

Review exponent notation and identify equivalent exponential expressions.

Evaluate numerical and algebraic expressions with exponents using the order of operations.

Investigate patterns of exponents with positive/negative bases and even/odd bases.

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Topic B: Properties of Exponents

Investigate exponent patterns to write equivalent expressions.

Apply the product of powers rule and the quotient of powers rule to write equivalent, simplified exponential expressions.

Apply the power of powers rule and power of product rule to write equivalent, simplified exponential expressions.

Reason with zero exponents to write equivalent, simplified exponential expressions.

Reason with negative exponents to write equivalent, simplified exponential expressions.

Topic C: Scientific Notation

Write large and small numbers as powers of 10. 

8.EE.A.3 8.EE.A.4

Define and write numbers in scientific notation.

Compare numbers written in scientific notation.

Multiply and divide with numbers in scientific notation. Interpret scientific notation on calculators.

Add and subtract with numbers in scientific notation.

Solve multi-step applications using scientific notation and properties of exponents.

8.EE.A.1 8.EE.A.3 8.EE.A.4

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Exponents and Scientific Notation Stations Activity | Properties of Exponents

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Description

These Exponents and Scientific Notation Stations consist of 10 stations where students practice using the properties of exponents and scientific notation.

This activity is great for in class review!  Students can get out of their seats and move around, while still applying their understanding of the properties of exponents and scientific notation.

Students are able to practice and apply concepts with this exponents and scientific notation activity, while collaborating and having fun! Math can be fun and interactive!

Standards: CCSS (8.EE.1, 8.EE.2, 8.EE.3, 8.EE.4) and TEKS (8.2.C)

More details on what is included:

10 Stations that can be utilized in pairs or groups of 3-4 and any necessary recording sheets and answer keys.

• 10 Stations: Exponents and Scientific Notation
• Recording Sheet
• Teacher Directions

***Please download a preview to see sample pages and more information.***

How to use this resource:

• Use as a whole group classroom activity
• Use in a small group for additional remediation, tutoring, or enrichment
• Use as an alternative homework or independent practice assignment
• Incorporate within our Exponents and Scientific Notation Unit to support the mastery of concepts and skills.

Time to Complete:

• Most activities can be utilized within one class period. Performance tasks summarize the entire unit and may need 2-3 class periods. However, feel free to review the activities and select specific problems to meet your students’ needs and time specifications. There are multiple problems to practice the same concepts, so you can adjust as needed.

Looking for instructional materials? 

• Check out the corresponding Exponents and Scientific Notation Unit , which includes student handouts, independent practice, assessments, and answer keys.

More 8th Grade Activity Bundles:

Unit 1: Real Number System 

Unit 2: Exponents and Scientific Notation

Unit 3: Linear Equations

Unit 4: Linear Relationships

Unit 5: Functions

Unit 6: Systems of Equations

Unit 7: Transformations

Unit 8: Angle Relationships

Unit 9: Pythagorean Theorem

Unit 10: Volume Unit 11: Scatter Plots and Data

More 8th Grade Units:

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Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation

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Go Math Grade 8 Chapter 2 Exponents and Scientific Notation Answer Key

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Lesson 1: Integer Exponents

• Integer Exponents – Page No. 36
• Integer Exponents – Page No. 37
• Integer Exponents Lesson Check – Page No. 38

Lesson 2: Scientific Notation with Positive Powers of 10

• Scientific Notation with Positive Powers of 10 – Page No. 42
• Scientific Notation with Positive Powers of 10 – Page No. 43
• Scientific Notation with Positive Powers of 10 Lesson Check – Page No. 44

Lesson 3: Scientific Notation with Negative Powers of 10

• Scientific Notation with Negative Powers of 10 – Page No. 48
• Scientific Notation with Negative Powers of 10 – Page No. 49
• Scientific Notation with Negative Powers of 10 Lesson Check – Page No. 50

Lesson 4: Operations with Scientific Notation

• Operations with Scientific Notation – Page No. 54
• Operations with Scientific Notation – Page No. 55
• Operations with Scientific Notation Lesson Check – Page No. 56
• Model Quiz – Page No. 57

Mixed Review

• Mixed Review – Page No. 58

Guided Practice – Integer Exponents – Page No. 36

Find the value of each power.

Question 1. 8 −1 = \(\frac{□}{□}\)

Answer: \(\frac{1}{8}\)

Explanation: Base = 8 Exponent = 1 8 −1 = (1/8) 1 = 1/8

Question 2. 6 −2 = \(\frac{□}{□}\)

Answer: \(\frac{1}{36}\)

Explanation: Base = 6 Exponent = 2 6 −2 = (1/6) 2 = 1/36

Exponents Grade 8 Worksheet Question 3. 256 0 = ______

Explanation: 256 0 Base = 256 Exponent = 0 Anything raised to the zeroth power is 1. 256 0 = 1

Question 4. 10 2 = ______

Answer: 100

Explanation: Base = 10 Exponent = 2 10 2 = 10 × 10 = 100

Question 5. 5 4 = ______

Answer: 625

Explanation: Base = 5 Exponent = 4 5 4 = 5 × 5 × 5 × 5 = 625

Question 6. 2 −5 = \(\frac{□}{□}\)

Answer: \(\frac{1}{32}\)

Explanation: Base = 2 Exponent = 5 2 −5 = (1/2) 5 = (1/2) × (1/2) × (1/2) × (1/2) × (1/2) = 1/32

Question 7. 4 −5 = \(\frac{□}{□}\)

Answer: \(\frac{1}{1,024}\)

Explanation: Base = 4 Exponent = 5 4 −5 = (1/4) 5 = (1/4) × (1/4) × (1/4) × (1/4) × (1/4) = 1/1,024

Question 8. 89 0 = ______

Explanation: 89 0 Base = 89 Exponent = 0 Anything raised to the zeroth power is 1. 89 0 = 1

Exponents and Scientific Notation Unit Test Answer Key 8th Grade Question 9. 11 −3 = \(\frac{□}{□}\)

Answer: \(\frac{1}{1,331}\)

Explanation: Base = 11 Exponent = 3 11 −3 = (1/11) 3 = (1/11) × (1/11) × (1/11) = 1/1,331

Use properties of exponents to write an equivalent expression.

Question 10. 4 ⋅ 4 ⋅ 4 = 4 ? Type below: _____________

Answer: 4 3

Explanation: The same number 4 is multiplying 3 times. The number of times a term is multiplied is called the exponent. So the base is 4 and the exponent is 3 4 ⋅ 4 ⋅ 4 = 4 3

Question 11. (2 ⋅ 2) ⋅ (2 ⋅ 2 ⋅ 2) = 2 ? ⋅ 2 ? = 2 ? Type below: _____________

Answer: 2 5

Explanation: The same number 2 is multiplying 5 times. The number of times a term is multiplied is called the exponent. So the base is 2 and the exponent is 5 (2 ⋅ 2) ⋅ (2 ⋅ 2 ⋅ 2) = 2 2 ⋅ 2 3 = 2 5

Question 12. \(\frac { { 6 }^{ 7 } }{ { 6 }^{ 5 } } \) = \(\frac{6⋅6⋅6⋅6⋅6⋅6⋅6}{6⋅6⋅6⋅6⋅6}\) = 6 ? Type below: _____________

Answer: 6 2

Explanation: \(\frac { { 6 }^{ 7 } }{ { 6 }^{ 5 } } \) = \(\frac{6⋅6⋅6⋅6⋅6⋅6⋅6}{6⋅6⋅6⋅6⋅6}\) Cancel the common factors 6.6 Base = 6 Exponent = 2 6 2

Question 13. \(\frac { { 8 }^{ 12 } }{ { 8 }^{ 9 } } \) = 8 ?-? = 8 ? Type below: _____________

Answer: 8 3

Explanation: \(\frac { { 8 }^{ 12 } }{ { 8 }^{ 9 } } \) Bases are common. So, the exponents are subtracted 8 12-9 = 8 3

Add and Subtract Scientific Notation Calculator Question 14. 5 10 ⋅ 5 ⋅ 5 = 5 ? Type below: _____________

Answer: 5 12

Explanation: Bases are common and multiplied. So, the exponents are added Base = 5 Exponents = 10 + 1 + 1 = 12 5 12

Question 15. 7 8 ⋅ 7 5 = 7 ? Type below: _____________

Answer: 7 13

Explanation: Bases are common and multiplied. So, the exponents are added Base = 7 Exponents = 8 + 5 = 13 7 13

Question 16. (6 2 ) 4 = (6 ⋅ 6) ? = (6 ⋅ 6) ⋅ (6 ⋅ 6) ⋅ (? ⋅ ?) ⋅ ? = 6 ? Type below: _____________

Answer: 6 8

Explanation: (6 2 ) 4 = (6 ⋅ 6) 4 = (6 ⋅ 6) ⋅ (6 ⋅ 6) ⋅ (6 ⋅ 6) ⋅ (6 ⋅ 6) = 6 2 ⋅ 6 2 . 6 2 ⋅ 6 2 Bases are common and multiplied. So, the exponents are added = 6 2+2+2+2 6 8

Question 17. (3 3 ) 3 = (3 ⋅ 3 ⋅ 3) 3 = (3 ⋅ 3 ⋅ 3) ⋅ (? ⋅ ? ⋅ ?) ⋅ ? = 3 ? Type below: ______________

Answer: 3 9

Explanation: (3 ⋅ 3 ⋅ 3) ⋅ (3 ⋅ 3 ⋅ 3) ⋅ (3 ⋅ 3 ⋅ 3) = 3 3 ⋅ 3 3 ⋅ 3 3 Bases are common and multiplied. So, the exponents are added 3 3 + 3 + 3 3 9

Simplify each expression.

Question 18. (10 − 6) 3 ⋅4 2 + (10 + 2) 2 ______

Answer: 1,168

Explanation: 4³. 4² + (12)² = 4 5 + (12)² = 4 5 + (12 . 12)² 4 5 + (144) = 1,024 + 144 = 1,168

Question 19. \(\frac { { (12-5) }^{ 7 } }{ { [(3+4)^{ 2 }] }^{ 2 } } \) ________

Answer: 343

Explanation: 7 7 ÷ (7²)² = 7 7 ÷ 7 4 7 7-4 7³ 7 . 7 . 7 = 343

ESSENTIAL QUESTION CHECK-IN

Question 20. Summarize the rules for multiplying powers with the same base, dividing powers with the same base, and raising a power to a power. Type below: ______________

Answer: The exponent “product rule” tells us that, when multiplying two powers that have the same base, you can add the exponents. The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. The “power rule” tells us that to raise a power to a power, just multiply the exponents.

Independent Practice – Integer Exponents – Page No. 37

Question 21. Explain why the exponents cannot be added in the product 12 3 ⋅ 11 3 . Type below: ______________

Answer: The exponent “product rule” tells us that, when multiplying two powers that have the same base, you can add the exponents. The bases are not the same in the given problem. => (12)³ x (11)³ If we solve this equation following the rule of exponent will get the correct answer: => (12 x 12 x 12) x (11 x 11 x 11) => 1728 X 1331 => The answer is 2 299 968 But if we add the exponent, the answer would be wrong => (12)³ x (11)³ => 132^6 => 5289852801024 which is wrong.

Question 22. List three ways to express 3 5 as a product of powers. Type below: ______________

Answer: 3¹ . 3 4 3² . 3 3 3³ . 3 2

Question 23. Astronomy The distance from Earth to the moon is about 22 4 miles. The distance from Earth to Neptune is about 22 7 miles. Which distance is the greater distance and about how many times greater is it? _______ times

Answer: (22)³ or 10,648 times

Explanation: The distance from Earth to the moon is about 22 4 miles. The distance from Earth to Neptune is about 22 7 miles. 22 7 – 22 4   = (22)³ The greatest distance is from Earth to Neptune The distance from Earth to Neptune is greater by (22)³ or 10,648 miles

Question 24. Critique Reasoning A student claims that 8 3 ⋅ 8 -5 is greater than 1. Explain whether the student is correct or not. ______________

Answer: 8 3 ⋅ 8 -5 is = 8 -2 (1/8)² (1/8) . (1/8) = 1/64 = 0.015 The student is not correct.

Find the missing exponent.

Question 25. (b 2 ) ? = b -6 _______

Answer: (b 2 ) -8

Explanation: (b 2 ) ? = b -6 (b -6 ) = b 2-8 (b 2-8 ) = b 2  .  b -8 (b 2 ) -8 = b -6

Grade 8 Lesson 1 Properties of Integer Exponents Quiz Answer Key Question 26. x ? ⋅ x 6 = x 9 _______

Explanation: x ? ⋅ x 6 = x 9 x 9 = x 3 + 6 x³ x 6

Question 27. \(\frac { { y }^{ 25 } }{ { y }^{ ? } } \) = y 6 _______

Answer: y25 ÷ y16

Explanation: \(\frac { { y }^{ 25 } }{ { y }^{ ? } } \) = y 6  y 6 = y 25 – 16 y 25 ÷ y 16

Question 28. Communicate Mathematical Ideas Why do you subtract exponents when dividing powers with the same base? Type below: ______________

Answer: To divide exponents (or powers) with the same base, subtract the exponents. The division is the opposite of multiplication, so it makes sense that because you add exponents when multiplying numbers with the same base, you subtract the exponents when dividing numbers with the same base.

Question 29. Astronomy The mass of the Sun is about 2 × 10 27 metric tons, or 2 × 10 30 kilograms. How many kilograms are in one metric ton? ________ kgs in one metric ton

Answer: 1,000 kgs in one metric ton

Explanation: The mass of the Sun is about 2 × 10 27 metric tons, or 2 × 10 30 kilograms. 2 × 10 27 metric tons = 2 × 10 30 ki 1 metric ton = 2 × 10 30 ki ÷ 2 × 10 27 = (10)³ = 1,000 kgs in one metric ton

Question 30. Represent Real-World Problems In computer technology, a kilobyte is 2 10 bytes in size. A gigabyte is 2 30 bytes in size. The size of a terabyte is the product of the size of a kilobyte and the size of a gigabyte. What is the size of a terabyte? Type below: ______________

Answer: 2 40  bytes

Explanation: In computer technology, a kilobyte is 2 10 bytes in size. A gigabyte is 2 30 bytes in size. The size of a terabyte is the product of the size of a kilobyte and the size of a gigabyte. terabyte = 2 10 bytes × 2 30 bytes = 2 10+30 bytes = 2 40  bytes

Integer Exponents – Page No. 38

Question 31. Write equivalent expressions for x 7 ⋅ x -2 and \(\frac { { x }^{ 7 } }{ { x }^{ 2 } } \). What do you notice? Explain how your results relate to the properties of integer exponents. Type below: ______________

Answer: x^a * x^b = x^(a+b) and x^-a = 1/x^a Therefore, x^7 * x^-2 = x^7/x^2 = x^5 or x^7 * x^-2 = x^(7-2) = x^5 x^7 / x^2 = x^7 * x^-2

Question 32. Look for a Pattern Describe any pattern you see in the table. Type below: ______________

Answer: As the number of rows increased, the number of cubes in each row by a multiple of 3.

Exponents Questions and Answers Grade 8 Question 33. Using exponents, how many cubes will be in Row 6? How many times as many cubes will be in Row 6 than in Row 3? _______ times more cubes

Answer: (3 3 ) times more cubes

Explanation: For row 6, the number of cubes in each row = (3 6 ) (3 6 ) ÷ (3 3 ) = (3 6-3 ) = (3 3 ) (3 3 ) times more cubes

Question 34. Justify Reasoning If there are 6 rows in the triangle, what is the total number of cubes in the triangle? Explain how you found your answer. ______ cubes

Answer: 1,092 cubes

Explanation: (3 1 ) + (3 2 ) + (3 3 ) + (3 4 ) + (3 5 ) + (3 6 ) 3 + 9 + 27 + 81 + 243 + 729 = 1,092

Focus on Higher Order Thinking

Question 35. Critique Reasoning A student simplified the expression \(\frac { { 6 }^{ 2 } }{ { 36 }^{ 2 } } \) as \(\frac{1}{3}\). Do you agree with this student? Explain why or why not. ______________

Answer: \(\frac { { 6 }^{ 2 } }{ { 36 }^{ 2 } } \) (6 2 ) ÷ (6 2 )² (6 2 ) ÷ (6 4 ) (6 2 – 4 ) (6 -2 ) = 1/36 I don’t agree with the student

Question 36. Draw Conclusions Evaluate –a n when a = 3 and n = 2, 3, 4, and 5. Now evaluate (–a) n when a = 3 and n = 2, 3, 4, and 5. Based on this sample, does it appear that –a n = (–a) n ? If not, state the relationships, if any, between –a n and (–a) n . Type below: ______________

Answer: –a n when a = 3 and n = 2, 3, 4, and 5. -3 n -(3 2 )= -9 (–a) n = -3 . -3 = 9 –a n = (–a) n are not equal.

Properties of Integer Exponents Worksheet Question 37. Persevere in Problem-Solving A number to the 12th power divided by the same number to the 9th power equals 125. What is the number? _______

Answer: Let’s call our number a. (a 12 ) ÷ (a 9  ) (a 12-9 ) = (a 3 ) (a 3 ) = 125 a = (125) 1/3 a = 5

Guided Practice – Scientific Notation with Positive Powers of 10 – Page No. 42

Write each number in scientific notation.

Question 1. 58,927 (Hint: Move the decimal left 4 places) Type below: ______________

Answer: 5.8927 × (10) 4

Explanation: 58,927 Move the decimal left 4 places 5.8927 × (10) 4

Scientific Notation Exercises Worksheet Question 2. 1,304,000,000 (Hint: Move the decimal left 9 places.) Type below: ______________

Answer: 1.304 × (10) 9

Explanation: 1,304,000,000 Move the decimal left 9 places 1.304 × (10) 9

Question 3. 6,730,000 Type below: ______________

Explanation: 6,730,000 Move the decimal left 6 places 6.73 × (10) 6

Question 4. 13,300 Type below: ______________

Explanation: 13,300 Move the decimal left 4 places 1.33 × (10) 4

Question 5. An ordinary quarter contains about 97,700,000,000,000,000,000,000 atoms. Type below: ______________

Explanation: 97,700,000,000,000,000,000,000 Move the decimal left 22 places 9.77 × (10) 22

Question 6. The distance from Earth to the Moon is about 384,000 kilometers. Type below: ______________

Answer: 3.84 × (10) 6

Explanation: 384,000 Move the decimal left 6 places 3.84 × (10) 6

Write each number in standard notation.

Question 7. 4 × 10 5 (Hint: Move the decimal right 5 places.) Type below: ______________

Answer: 400,000

Explanation: 4 × 10 5 Move the decimal right 5 places 400,000

Lesson 2 Problem Set 2.1 Answer Key Question 8. 1.8499 × 10 9 (Hint: Move the decimal right 9 places.) Type below: ______________

Answer: 1849900000

Explanation: 1.8499 × 10 9 Move the decimal right 9 places 1849900000

Question 9. 6.41 × 10 3 Type below: ______________

Answer: 6410

Explanation: 6.41 × 10 3 Move the decimal right 3 places 6410

Question 10. 8.456 × 10 7 Type below: ______________

Answer: 84560000

Explanation: 8.456 × 10 7 Move the decimal right 7 places 84560000

Question 11. 8 × 10 5 Type below: ______________

Answer: 800,000

Explanation: 8 × 10 5 Move the decimal right 5 places 800,000

Question 12. 9 × 10 10 Type below: ______________

Answer: 90000000000

Explanation: 9 × 10 10 Move the decimal right 10 places 90000000000

Scientific Notation Worksheet 8th Grade Pdf Question 13. Diana calculated that she spent about 5.4 × 10 4 seconds doing her math homework during October. Write this time in standard notation. Type below: ______________

Answer: 5400

Explanation: Diana calculated that she spent about 5.4 × 10 4 seconds doing her math homework during October. 5.4 × 10 4 Move the decimal right 4 places 5400

Question 14. The town recycled 7.6 × 10 6 cans this year. Write the number of cans in standard notation Type below: ______________

Answer: 7600000

Explanation: The town recycled 7.6 × 10 6 cans this year. 7.6 × 10 6 Move the decimal right 10 places 7600000

Question 15. Describe how to write 3,482,000,000 in scientific notation. Type below: ______________

Answer: 3.482 × (10) 9

Explanation: 3,482,000,000 Move the decimal left 9 places 3.482 × (10) 9

Independent Practice – Scientific Notation with Positive Powers of 10 – Page No. 43

Paleontology

Question 16. Apatosaurus ______________ Type below: ______________

Answer: 6.6 × (10) 4

Explanation: 66,000 Move the decimal left 4 places 6.6 × (10) 4

Question 17. Argentinosaurus ___________ Type below: ______________

Answer: 2.2 × (10) 5

Explanation: 220,000 Move the decimal left 5 places 2.2 × (10) 5

Question 18. Brachiosaurus ______________ Type below: ______________

Answer: 1 × (10) 5

Explanation: 100,000 Move the decimal left 5 places 1 × (10) 5

Lesson 2 Extra Practice Powers and Exponents Answer Key Question 19. Camarasaurus ______________ Type below: ______________

Answer: 4 × (10) 4

Explanation: 40,000 Move the decimal left 4 places 4 × (10) 4

Question 20. Cetiosauriscus ____________ Type below: ______________

Answer: 1.985 × (10) 4

Explanation: 19,850 Move the decimal left 4 places 1.985 × (10) 4

Question 21. Diplodocus _____________ Type below: ______________

Answer: 5 × (10) 4

Explanation: 50,000 Move the decimal left 4 places 5 × (10) 4

Question 22. A single little brown bat can eat up to 1,000 mosquitoes in a single hour. Express in scientific notation how many mosquitoes a little brown bat might eat in 10.5 hours. Type below: ______________

Answer: 1.05 × (10) 4

Explanation: (1000 x 10.5) = 10500. The little brown bat can eat 10500 mosquitoes in 10.5 hours. 1.05 × (10) 4

Question 23. Multistep Samuel can type nearly 40 words per minute. Use this information to find the number of hours it would take him to type 2.6 × 10 5 words. Type below: ______________

Answer: Samuel can type 40 words per minute. Then how many hours will it take for him to type 2.6 words times 10 to the power of five words 2.6 words time 10 to the power of 5 2.6 × (10) 4 2.6 x 100 000 = 260 000 words in all. Now, we need to find the number of words Samuel can type in an hour 40 words/minutes, in 1 hour there are 60 minutes 40 x 60 2,400 words /hour Now, let’s divide the total of words he needs to type to the number of words he can type in an hour 260 000 / 2 400 108.33 hours.

Question 24. Entomology A tropical species of mite named Archegozetes longisetosus is the record holder for the strongest insect in the world. It can lift up to 1.182 × 10 3 times its own weight. a. If you were as strong as this insect, explain how you could find how many pounds you could lift. Type below: ______________

Answer: Number of pounds you can lift by 1.182 × 10 3 by your weight

Question 24. b. Complete the calculation to find how much you could lift, in pounds, if you were as strong as an Archegozetes longisetosus mite. Express your answer in both scientific notation and standard notation. Type below: ______________

Answer: scientific notation: 1.182 × 10 5 standard notation: 118200

Explanation: 1.182 × 10 3 × 10 2 1.182 × 10 5 118200

Question 25. During a discussion in science class, Sharon learns that at birth an elephant weighs around 230 pounds. In four herds of elephants tracked by conservationists, about 20 calves were born during the summer. In scientific notation, express approximately how much the calves weighed all together. Type below: ______________

Answer: 4.6 × 10 3

Explanation: During a discussion in science class, Sharon learns that at birth an elephant weighs around 230 pounds. In four herds of elephants tracked by conservationists, about 20 calves were born during the summer. Total weight of the claves = 230 × 20 = 4600 Move the decimal left 3 places 4.6 × 10 3

Question 26. Classifying Numbers Which of the following numbers are written in scientific notation? 0.641 × 10 3           9.999 × 10 4 2 × 10 1                  4.38 × 5 10 Type below: ______________

Answer: 0.641 × 10 3 4.38 × 5 10

Scientific Notation with Positive Powers of 10 – Page No. 44

Question 27. Explain the Error Polly’s parents’ car weighs about 3500 pounds. Samantha, Esther, and Polly each wrote the weight of the car in scientific notation. Polly wrote 35.0 × 10 2 , Samantha wrote 0.35 × 10 4 , and Esther wrote 3.5 × 10 4 . a. Which of these girls, if any, is correct? ______________

Answer: None of the girls is correct

Question 27. b. Explain the mistakes of those who got the question wrong. Type below: ______________

Answer: Polly did not express the number such first part is greater than or equal to 1 and less than 10 Samantha did not express the number such first part is greater than or equal to 1 and less than 10 Esther did not express the exponent of 10 correctly

Question 28. Justify Reasoning If you were a biologist counting very large numbers of cells as part of your research, give several reasons why you might prefer to record your cell counts in scientific notation instead of standard notation. Type below: ______________

Answer: It is easier to comprehend the magnitude of large numbers when in scientific notation as multiple zeros in the number are removed and express as an exponent of 10. It is easier to compare large numbers when in scientific notation as numbers are be expressed as a product of a number greater than or equal to 1 and less than 10 It is easier to multiply the numbers in scientific notation.

Question 29. Draw Conclusions Which measurement would be least likely to be written in scientific notation: number of stars in a galaxy, number of grains of sand on a beach, speed of a car, or population of a country? Explain your reasoning. Type below: ______________

Answer: speed of a car

Explanation: As we know scientific notation is used to express measurements that are extremely large or extremely small. The first two are extremely large, then, they could be expressed in scientific notation. If we compare the speed of a car and the population of a country, it is clear that the larger will be the population of a country. Therefore, it is more likely to express that in scientific notation, so the answer is the speed of a car.

Question 30. Analyze Relationships Compare the two numbers to find which is greater. Explain how you can compare them without writing them in standard notation first. 4.5 × 10 6               2.1 × 10 8 Type below: ______________

Answer: 2.1 × 10 8

Explanation: 2.1 × 10 8 is greater because the power of 10 is greater in  2.1 × 10 8

Question 31. Communicate Mathematical Ideas To determine whether a number is written in scientific notation, what test can you apply to the first factor, and what test can you apply to the second factor? Type below: ______________

Answer: The first term must have one number before the decimal point The second term (factor) must be 10 having some power.

Guided Practice – Scientific Notation with Negative Powers of 10 – Page No. 48

Question 1. 0.000487 Hint: Move the decimal right 4 places. Type below: ______________

Answer: 4.87 × 10 -4

Explanation: 0.000487 Move the decimal right 4 places 4.87 × 10 -4

How to Multiply Scientific Notation with Negative Exponents Question 2. 0.000028 Hint: Move the decimal right 5 places Type below: ______________

Answer: 2.8 × 10 -5

Explanation: 0.000028 Move the decimal right 5 places 2.8 × 10 -5

Question 3. 0.000059 Type below: ______________

Answer: 5.9 × 10 -5

Explanation: 0.000059 Move the decimal right 5 places 5.9 × 10 -5

Question 4. 0.0417 Type below: ______________

Answer: 4.17 × 10 -2

Explanation: 0.0417 Move the decimal right 2 places 4.17 × 10 -2

Question 5. Picoplankton can be as small as 0.00002 centimeters. Type below: ______________

Answer: 2 × 10 -5

Explanation: 0.00002 Move the decimal right 5 places 2 × 10 -5

Question 6. The average mass of a grain of sand on a beach is about 0.000015 grams. Type below: ______________

Answer: 1.5 × 10 -5

Explanation: 0.000015 Move the decimal right 5 places 1.5 × 10 -5

Question 7. 2 × 10 -5 Hint: Move the decimal left 5 places. Type below: ______________

Answer: 0.00002

Explanation: 2 × 10 -5 Move the decimal left 5 places 0.00002

Question 8. 3.582 × 10 -6 Hint: Move the decimal left 6 places. Type below: ______________

Answer: 0.000003582

Explanation: 3.582 × 10 -6 Move the decimal left 6 places 0.000003582

Question 9. 8.3 × 10 -4 Type below: ______________

Answer: 0.00083

Explanation: 8.3 × 10 -4 Move the decimal left 4 places 0.00083

Question 10. 2.97 × 10 -2 Type below: ______________

Answer: 0.0297

Explanation: 2.97 × 10 -2 Move the decimal left 2 places 0.0297

Question 11. 9.06 × 10 -5 Type below: ______________

Answer: 0.0000906

Explanation: 9.06 × 10 -5 Move the decimal left 5 places 0.0000906

Question 12. 4 × 10 -5 Type below: ______________

Answer: 0.00004

Explanation: 4 × 10 -5 Move the decimal left 5 places 0.00004

Question 13. The average length of a dust mite is approximately 0.0001 meters. Write this number in scientific notation. Type below: ______________

Answer: 1 × 10 -4

Explanation: The average length of a dust mite is approximately 0.0001 meters. 0.0001 Move the decimal right 4 places 1 × 10 -4

Question 14. The mass of a proton is about 1.7 × 10 -24 grams. Write this number in standard notation. Type below: ______________

Answer: 0.000000000000000000000017

Explanation: The mass of a proton is about 1.7 × 10 -24 grams. 1.7 × 10 -24 Move the decimal left 24 places 0.000000000000000000000017

Question 15. Describe how to write 0.0000672 in scientific notation. Type below: ______________

Answer: 6.72 × 10 -5

Explanation: 0.0000672 Move the decimal right 5 places 6.72 × 10 -5

Independent Practice – Scientific Notation with Negative Powers of 10 – Page No. 49

Question 16. Alpaca _______ Type below: ______________

Answer: 2.77 × 10 -3

Explanation: 0.00277 Move the decimal right 3 places 2.77 × 10 -3

Question 17. Angora rabbit _____________ Type below: ______________

Answer: 1.3 × 10 -3

Explanation: 0.0013 Move the decimal right 3 places 1.3 × 10 -3

Question 18. Llama ____________ Type below: ______________

Answer: 3.5 × 10 -3

Explanation: 0.0035 Move the decimal right 3 places 3.5 × 10 -3

Question 19. Angora goat ____________ Type below: ______________

Answer: 4.5 × 10 -3

Explanation: 0.0045 Move the decimal right 3 places 4.5 × 10 -3

Question 20. Orb web spider ___________ Type below: ______________

Answer: 1.5 × 10 -2

Explanation: 0.015 Move the decimal right 2 places 1.5 × 10 -2

Question 21. Vicuña __________ Type below: ______________

Answer: 8 × 10 -4

Explanation: 0.0008 Move the decimal right 4 places 8 × 10 -4

Question 22. Make a Conjecture Which measurement would be least likely to be written in scientific notation: the thickness of a dog hair, the radius of a period on this page, the ounces in a cup of milk? Explain your reasoning. Type below: ______________

Answer: The ounces in a cup of milk would be least likely to be written in scientific notation. The ounces in a cup of milk is correct. Scientific notation is used for either very large or extremely small numbers. The thickness of dog hair is very small as the hair is thin. Hence can be converted to scientific notation. The radius of a period on this page is also pretty small. Hence can be converted to scientific notation. The ounces in a cup of milk. There are 8 ounces in a cup, so this is least likely to be written in scientific notation.

Question 23. Multiple Representations Convert the length 7 centimeters to meters. Compare the numerical values when both numbers are written in scientific notation Type below: ______________

Answer: 7 centimeters convert to meters In every 1 meter, there are 100 centimeters = 7/100 = 0.07 Therefore, in 7 centimeters there are 0.07 meters. 7 cm is a whole number while 0.07 m is a decimal number Scientific Notation of each number 7 cm = 7 x 10° 7 m = 1 x 10¯² Scientific notation, by the way, is an expression used by the scientist to make a large number of very small number easy to handle.

Question 24. Draw Conclusions A graphing calculator displays 1.89 × 10 12 as 1.89E12. How do you think it would display 1.89 × 10 -12 ? What does the E stand for? Type below: ______________

Answer: 1.89E-12. E= Exponent

Explanation:

Question 25. Communicate Mathematical Ideas When a number is written in scientific notation, how can you tell right away whether or not it is greater than or equal to 1? Type below: ______________

Answer: A number written in scientific notation is of the form a × 10 -n  where 1 ≤ a < 10 and n is an integer The number is greater than or equal to one if n ≥ 0.

Question 26. The volume of a drop of a certain liquid is 0.000047 liter. Write the volume of the drop of liquid in scientific notation. Type below: ______________

Answer: 4.7 × 10 -5

Explanation: The volume of a drop of a certain liquid is 0.000047 liter. Move the decimal right 5 places 4.7 × 10 -5

Question 27. Justify Reasoning If you were asked to express the weight in ounces of a ladybug in scientific notation, would the exponent of the 10 be positive or negative? Justify your response. ______________

Answer: Negative

Explanation: Scientific notation is used to express very small or very large numbers. Very small numbers are written in scientific notation using negative exponents. Very large numbers are written in scientific notation using positive exponents. Since a ladybug is very small, we would use the very small scientific notation, which uses negative exponents.

Physical Science – Scientific Notation with Negative Powers of 10 – Page No. 50

Question 28. Type below: ______________

Answer: 1.74 × (10) 6

Explanation: The moon = 1,740,000 Move the decimal left 6 places 1.74 × (10) 6

Question 29. Type below: ______________

Answer: 1.25e-10

Explanation: 1.25 × (10) -10 Move the decimal left 10 places 1.25e-10

Question 30. Type below: ______________

Answer: 2.8 × (10) 3

Explanation: 0.0028 Move the decimal left 3 places 2.8 × (10) 3

Question 31. Type below: ______________

Answer: 71490000

Explanation: 7.149 × (10) 7 Move the decimal left 7 places 71490000 Question 32. Type below: ______________

Answer: 1.82 × (10) -10

Explanation: 0.000000000182 Move the decimal right 10 places 1.82 × (10) -10

Question 33. Type below: ______________

Answer: 3397000

Explanation: 3.397 × (10) 6 Move the decimal left 6 places 3397000

Question 34. List the items in the table in order from the smallest to the largest. Type below: ______________

Answer: 1.82 × (10) -10 1.25 × (10) -10 2.8 × (10) 3 1.74 × (10) 6 3.397 × (10) 6 7.149 × (10) 7

Question 35. Analyze Relationships Write the following diameters from least to greatest. 1.5 × 10 -2 m ; 1.2 × 10 2 m ; 5.85 × 10 -3 m ; 2.3 × 10 -2 m ; 9.6 × 10 -1 m. Type below: ______________

Answer: 5.85 × 10 -3 m, 1.5 × 10 -2 m, 2.3 × 10 -2 m, 9.6 × 10 -1 m, 1.2 × 10 2 m

Explanation: 1.5 × 10 -2 m = 0.015 1.2 × 10 2 m = 120 5.85 × 10 -3 m = 0.00585 2.3 × 10 -2 m = 0.023 9.6 × 10 -1 m = 0.96 0.00585, 0.015, 0.023, 0.96, 120

Question 36. Critique Reasoning Jerod’s friend Al had the following homework problem: Express 5.6 × 10 -7 in standard form. Al wrote 56,000,000. How can Jerod explain Al’s error and how to correct it? Type below: ______________

Explanation: 5.6 × 10 -7 in 0.000000056 Al wrote 56,000,000. AI wrote the zeroes to the right side of the 56 which is not correct. As the exponent of 10 is negative zero’s need to add to the left of the number.

Question 37. Make a Conjecture Two numbers are written in scientific notation. The number with a positive exponent is divided by the number with a negative exponent. Describe the result. Explain your answer. Type below: ______________

Answer: When the division is performed, the denominator exponent is subtracted from the numerator exponent. Subtracting a negative value from the numerator exponent will increase its value.

Guided Practice – Operations with Scientific Notation – Page No. 54

Add or subtract. Write your answer in scientific notation.

Question 1. 4.2 × 10 6 + 2.25 × 10 5 + 2.8 × 10 6 4.2 × 10 6 + ? × 10 ? + 2.8 × 10 6 4.2 + ? + ? ? × 10? Type below: ______________

Answer: 4.2 × 10 6 + 0.225 × 10 × 10 5 + 2.8 × 10 6 Rewrite 2.25 = 0.225 × 10 (4.2 + 0.225 + 2.8) × 10 6 7.225 × 10 6

Question 2. 8.5 × 10 3 − 5.3 × 10 3 − 1.0 × 10 2 8.5 × 10 3 − 5.3 × 10 3 − ? × 10? ? − ? − ? ? × 10 ? Type below: ______________

Answer: 8.5 × 10 3 − 5.3 × 10 3 − 0.1 × 10 3 (8.5 − 5.3 − 0.1) × 10 3 (3.1) × 10 3

Lesson 2 Multiplication of Numbers in Exponential Form Answer Key Question 3. 1.25 × 10 2 + 0.50 × 10 2 + 3.25 × 10 2 Type below: ______________

Answer: 1.25 × 10 2 + 0.50 × 10 2 + 3.25 × 10 2 (1.25 + 0.50 + 3.25) × 10 2 5 × 10 2

Question 4. 6.2 × 10 5 − 2.6 × 10 4 − 1.9 × 10 2 Type below: ______________

Answer: 6.2 × 10 5 − 2.6 × 10 4 − 1.9 × 10 2 6.2 × 10 5 − 0.26 × 10 5 − 0.0019 × 10 5 (6.2 – 0.26 – 0.0019) × 10 5 5.9381 × 10 5

Multiply or divide. Write your answer in scientific notation.

Question 5. (1.8 × 10 9 )(6.7 × 10 12 ) Type below: ______________

Answer: 12.06 × 10 21

Explanation: (1.8 × 10 9 )(6.7 × 10 12 ) 1.8 × 6.7 = 12.06 10 9+12  = 10 21 12.06 × 10 21

Question 6. \(\frac { { 3.46×10 }^{ 17 } }{ { 2×10 }^{ 9 } } \) Type below: ______________

Answer: 1.73 × 10 8

Explanation: 3.46/2 = 1.73 10 17 /10 9  = 10 17-9  = 10 8 1.73 × 10 8

Question 7. (5 × 10 12 )(3.38 × 10 6 ) Type below: ______________

Answer: 16.9 × 10 18

Explanation: (5 × 10 12 )(3.38 × 10 6 ) 5 × 3.38 = 16.9 10 6+12 = 10 18 16.9 × 10 18

Question 8. \(\frac { { 8.4×10 }^{ 21 } }{ { 4.2×10 }^{ 14 } } \) Type below: ______________

Answer: 2 × 10 7

Explanation: 8.4/4.2 = 2 10 21 /10 14 = 10 21-14 = 10 7 2 × 10 7

Write each number using calculator notation.

Question 9. 3.6 × 10 11 Type below: ______________

Answer: 3.6e11

Question 10. 7.25 × 10 -5 Type below: ______________

Answer: 7.25e-5

Question 11. 8 × 10 -1 Type below: ______________

Answer: 8e-1

Write each number using scientific notation.

Question 12. 7.6E − 4 Type below: ______________

Answer: 7.6 × 10 -4

Question 13. 1.2E16 Type below: ______________

Answer: 1.2 × 10 16

Question 14. 9E1 Type below: ______________

Answer: 9 × 10 1

Question 15. How do you add, subtract, multiply, and divide numbers written in scientific notation? Type below: ______________

Answer: Numbers with exponents can be added and subtracted only when they have the same base and exponent. To multiply two numbers in scientific notation, multiply their coefficients and add their exponents. To divide two numbers in scientific notation, divide their coefficients, and subtract their exponents.

Independent Practice – Operations with Scientific Notation – Page No. 55

Question 16. An adult blue whale can eat 4.0 × 10 7 krill in a day. At that rate, how many krill can an adult blue whale eat in 3.65 × 10 2 days? Type below: ______________

Answer: 14.6 × 10 9

Explanation: (4.0 × 10 7 )(3.65 × 10 2 ) 4.0 × 3.65 = 14.6 10 7+2   =  10 9 14.6 × 10 9

How to Multiply and Divide in Scientific Notation Question 17. A newborn baby has about 26,000,000,000 cells. An adult has about 4.94 × 10 13 cells. How many times as many cells does an adult have as a newborn? Write your answer in scientific notation. Type below: ______________

Answer: 1.9 × 10 3

Explanation: 26,000,000,000 = 2.6 × 10 10 4.94 × 10 13 (4.94 × 10 13 )/(2.6 × 10 10  ) 1.9 × 10 3

Represent Real-World Problems

Question 18. What is the total amount of paper, glass, and plastic waste generated? Type below: ______________

Answer: 11.388 × 10 7

Explanation: 7.131 × 10 7  + 1.153 × 10 7  + 3.104 × 10 7 11.388 × 10 7

Question 19. What is the total amount of paper, glass, and plastic waste recovered? Type below: ______________

Answer: 5.025 × 10 7

Explanation: 4.457 × 10 7   + 0.313 × 10 7   + 0.255 × 10 7 5.025 × 10 7

How to Multiply Scientific Notation Question 20. What is the total amount of paper, glass, and plastic waste not recovered? Type below: ______________

Answer: 6.363 × 10 7

Explanation: (11.388 × 10 7 )  – (5.025 × 10 7 ) 6.363 × 10 7

Question 21. Which type of waste has the lowest recovery ratio? Type below: ______________

Answer: Plastics

Explanation: 7.131 × 10 7   – 4.457 × 10 7 = 2.674 × 10 7 1.153 × 10 7   – 0.313 × 10 7 = 0.84 × 10 7 3.104 × 10 7   – 0.255 × 10 7 = 2.849 × 10 7 Plastics have the lowest recovery ratio

Social Studies

Question 22. How many more people live in France than in Australia? Type below: ______________

Answer: 4.33 × 10 7

Explanation: (6.48 × 10 7 )  – (2.15× 10 7 ) 4.33 × 10 7

Question 23. The area of Australia is 2.95 × 10 6 square miles. What is the approximate average number of people per square mile in Australia? Type below: ______________

Answer: About 7 people per square mile

Explanation: 2.95 × 10 6 square miles = (2.15× 10 7 ) 1 square mile = (2.15× 10 7 )/(2.95 × 10 6 ) = 7.288

Question 24. How many times greater is the population of China than the population of France? Write your answer in standard notation. Type below: ______________

Answer: 20.52; there are about 20 people in China for every 1 person in France.

Multiplication of Numbers in Exponential Form Answer Key Question 25. Mia is 7.01568 × 10 6 minutes old. Convert her age to more appropriate units using years, months, and days. Assume each month to have 30.5 days. Type below: ______________

Answer: 13 years 3 months 22.5 days

Explanation: 7.01568 × 10 6 minutes (7.01568 × 10 6 minutes) ÷ (6 × 10 1 )(2.4 × 10 1 )(1.2 × 10 1 )(3.05 × 10 1 ) = (1.331 × 10 1 ) = 13 years 3 months 22.5 days

Operations with Scientific Notation – Page No. 56

Question 26. Courtney takes 2.4 × 10 4 steps during her long-distance run. Each step covers an average of 810 mm. What total distance (in mm) did Courtney cover during her run? Write your answer in scientific notation. Then convert the distance to the more appropriate unit kilometers. Write that answer in standard form. ______ km

Answer: 19.4 km

Explanation: Courtney takes 2.4 × 10 4 steps during her long-distance run. Each step covers an average of 810 mm. (2.4 × 10 4 steps) × 810mm (2.4 × 10 4 ) × (8.1 × 10 2 ) The total distance covered = (19.44 × 10 6 ) Convert to unit kilometers: (19.44 × 10 6 ) × (1 × 10 -6 ) (1.94 × 10 1 ) 19.4 km

Question 27. Social Studies The U.S. public debt as of October 2010 was $9.06 × 10 12 . What was the average U.S. public debt per American if the population in 2010 was 3.08 × 10 8 people? $ _______

Answer: $29,400 per American

Explanation: ($9.06 × 10 12 .)/(3.08 × 10 8 ) ($2.94 × 10 4 .) = $29,400 per American

Question 28. Communicate Mathematical Ideas How is multiplying and dividing numbers in scientific notation different from adding and subtracting numbers in scientific notation? Type below: ______________

Answer: When you multiply or divide in scientific notation, you just add or subtract the exponents. When you add or subtract in scientific notation, you have to make the exponents the same before you can do anything else.

Question 29. Explain the Error A student found the product of 8 × 10 6 and 5 × 10 9 to be 4 × 10 15 . What is the error? What is the correct product? Type below: ______________

Answer: The error the student makes is he multiplies the terms instead of the addition.

Explanation: product of 8 × 10 6 and 5 × 10 9 40 × 10 15 4 × 10 16 The student missed the 10 while multiplying the product of 8 × 10 6 and 5 × 10 9

Question 30. Communicate Mathematical Ideas Describe a procedure that can be used to simplify \(\frac { { (4.87×10 }^{ 12 }) – { (7×10 }^{ 10 }) }{ { (3×10 }^{ 7 })-{ (6.1×10 }^{ 8 }) } \). Write the expression in scientific notation in simplified form. Type below: ______________

Answer: \(\frac { { (4.87×10 }^{ 12 }) – { (7×10 }^{ 10 }) }{ { (3×10 }^{ 7 })-{ (6.1×10 }^{ 8 }) } \) \(\frac { { (487×10 }^{ 10 }) – { (7×10 }^{ 10 }) }{ { (3×10 }^{ 7 })-{ (61×10 }^{ 7 }) } \) (480 × 10 10  )/(64 × 10 7  ) 7.50 × 10³

2.1 Integer Exponents – Model Quiz – Page No. 57

Question 1. 3 -4 \(\frac{□}{□}\)

Answer: \(\frac{1}{81}\)

Explanation: Base = 3 Exponent = 4 3 -4 = (1/3) 4 = 1/81

Question 2. 35 0 ______

Explanation: 35 0 Base = 35 Exponent = 0 Anything raised to the zeroth power is 1. 35 0 = 1

Question 3. 4 4 ______

Answer: 256

Explanation: Base = 4 Exponent = 4 4 4 = 4 . 4 . 4 . 4 = 2561

Use the properties of exponents to write an equivalent expression.

Question 4. 8 3 ⋅ 8 7 Type below: ____________

Answer: 8 10

Explanation: 8 3 ⋅ 8 7 8 3+7 8 10

Question 5. \(\frac { 12^{ 6 } }{ 12^{ 2 } } \) Type below: ____________

Answer: 12 4

Explanation: 12 6 ÷ 12 2 12 6-2 12 4

Question 6. (10 3 ) 5 Type below: ____________

Answer: 10 8

Explanation: (10 3 ) 5 (10 3+5 ) (10 8 )

2.2 Scientific Notation with Positive Powers of 10

Convert each number to scientific notation or standard notation.

Question 7. 2,000 Type below: ____________

Answer: 2 × (10 3 )

Explanation: 2 × 1,000 Move the decimal left 3 places 2 × (10 3 )

Question 8. 91,007,500 Type below: ____________

Answer: 9.10075 × (10 7 )

Explanation: 91,007,500 Move the decimal left 7 places 9.10075 × (10 7 )

Question 9. 1.0395 × 10 9 Type below: ____________

Answer: 1039500000

Explanation: 1.0395 × 10 9 Move the decimal right 9 places 1039500000

Question 10. 4 × 10 2 Type below: ____________

Answer: 400

Explanation: 4 × 10 2 Move the decimal right 2 places 400

2.3 Scientific Notation with Negative Powers of 10

Question 11. 0.02 Type below: ____________

Answer: 2 × 10 -2

Explanation: 0.02 Move the decimal right 2 places 2 × 10 -2

Practice and Homework Lesson 2.3 Answer Key Question 12. 0.000701 Type below: ____________

Answer: 7.01 × 10 -4

Explanation: 0.000701 Move the decimal right 4 places 7.01 × 10 -4

Question 13. 8.9 × 10 -5 Type below: ____________

Answer: 0.000089

Explanation: 8.9 × 10 -5 Move the decimal left 5 places 0.000089

Question 14. 4.41 × 10 -2 Type below: ____________

Answer: 0.0441

Explanation: 4.41 × 10 -2 Move the decimal left 2 places 0.0441

2.4 Operations with Scientific Notation

Perform the operation. Write your answer in scientific notation.

Question 15. 7 × 10 6 − 5.3 × 10 6 Type below: ____________

Answer: 1.7 × 10 6

Explanation: 7 × 10 6 − 5.3 × 10 6 (7 – 5.3) × 10 6 1.7 × 10 6

Question 16. 3.4 × 10 4 + 7.1 × 10 5 Type below: ____________

Answer: 7.44 × 10 4

Explanation: 3.4 × 10 4 + 7.1 × 10 5 0.34 × 10 5 + 7.1 × 10 5 (0.34 + 7.1) × 10 5 7.44 × 10 5

Question 17. (2 × 10 4 )(5.4 × 10 6 ) Type below: ____________

Answer: 10.8 × 10 10

Explanation: (2 × 10 4 )(5.4 × 10 6 ) (2 × 5.4)(10 4 × 10 6 ) 10.8 × 10 10

Question 18. \(\frac { 7.86×10^{ 9 } }{ 3×10^{ 4 } } \) Type below: ____________

Answer: 2.62 × 10 5

Explanation: 7.86/3 = 2.62 10 9 /10 4  = 10 5 2.62 × 10 5

Question 19. Neptune’s average distance from the Sun is 4.503×10 9 km. Mercury’s average distance from the Sun is 5.791 × 10 7 km. About how many times farther from the Sun is Neptune than Mercury? Write your answer in scientific notation. Type below: ____________

Answer: (0.7776 × 10 2 km) = 77.76 times

Explanation: As Neptune’s average distance from the sun is 4.503×10 9 km and Mercury’s is 5.791 × 10 7 km (4.503×10 9 km)/(5.791 × 10 7 km) (0.7776 × 10 9-7 km) (0.7776 × 10 2 km) 77.76  times

Essential Question

Question 20. How is scientific notation used in the real world? Type below: ____________

Answer: Scientific notation is used to write very large or very small numbers using fewer digits.

Selected Response – Mixed Review – Page No. 58

Question 1. Which of the following is equivalent to 6 -3 ? Options: a. 216 b. \(\frac{1}{216}\) c. −\(\frac{1}{216}\) d. -216

Answer: b. \(\frac{1}{216}\)

Explanation: Base = 6 Exponent = 3 6 3 = (1/6) 3 = 1/216

Question 2. About 786,700,000 passengers traveled by plane in the United States in 2010. What is this number written in scientific notation? Options: a. 7,867 × 10 5 passengers b. 7.867 × 10 2 passengers c. 7.867 × 10 8 passengers d. 7.867 × 10 9 passengers

Answer: c. 7.867 × 10 8 passengers

Explanation: 786,700,000 Move the decimal left 8 places 7.867 × 10 8 passengers

Question 3. In 2011, the population of Mali was about 1.584 × 10 7 people. What is this number written in standard notation? Options: a. 1.584 people b. 1,584 people c. 15,840,000 people d. 158,400,000 people

Answer: c. 15,840,000 people

Explanation: 1.584 × 10 7 Move the decimal right 7 places 15,840,000 people

Question 4. The square root of a number is between 7 and 8. Which could be the number? Options: a. 72 b. 83 c. 51 d. 66

Answer: c. 51

Explanation: 7²= 49 8²=64 (49+64)/2 56.5

Question 5. Each entry-level account executive in a large company makes an annual salary of $3.48 × 10 4 . If there are 5.2 × 10 2 account executives in the company, how much do they make in all? Options: a. $6.69 × 10 1 b. $3.428 × 10 4 c. $3.532 × 10 4 d. $1.8096 × 10 7

Answer: d. $1.8096 × 10 7

Explanation: Each entry-level account executive in a large company makes an annual salary of $3.48 × 10 4 . If there are 5.2 × 10 2 account executives in the company, ($3.48 × 10 4 )( 5.2 × 10 2 ) $1.8096 × 10 7

Question 6. Place the numbers in order from least to greatest. 0.24,4 × 10 -2 , 0.042, 2 × 10 -4 , 0.004 Options: a. 2 × 10 -4 , 4 × 10 -2 , 0.004, 0.042, 0.24 b. 0.004, 2 × 10 -4 , 0.042, 4 × 10 -2 , 0.24 c. 0.004, 2 × 10 -4 , 4 × 10 -2 , 0.042, 0.24 d. 2 × 10 -4 , 0.004, 4 × 10 -2 , 0.042, 0.24

Answer: d. 2 × 10 -4 , 0.004, 4 × 10 -2 , 0.042, 0.24

Explanation: 2 × 10 -4  = 0.0002 4 × 10 -2  = 0.04

Question 7. Guillermo is 5 \(\frac{5}{6}\) feet tall. What is this number of feet written as a decimal? Options: a. 5.7 feet b. 5.\(\bar{7}\) feet c. 5.83 feet d. 5.8\(\bar{3}\) feet

Answer: c. 5.83 feet

Question 8. A human hair has a width of about 6.5 × 10 -5 meters. What is this width written in standard notation? Options: a. 0.00000065 meter b. 0.0000065 meter c. 0.000065 meter d. 0.00065 meter

Answer: c. 0.000065 meter

Explanation: 6.5 × 10 -5 meter = 0.000065

Question 9. Consider the following numbers: 7000, 700, 70, 0.7, 0.07, 0.007 a. Write the numbers in scientific notation. Type below: _____________

Answer: 7000 = 7 × 10³ 700 = 7 × 10² 70 = 7 × 10¹ 0.7 = 7 × 10¯¹ 0.07 = 7 × 10¯² 0.007 = 7 × 10¯³

Question 9. b. Look for a pattern in the given list and the list in scientific notation. Which numbers are missing from the lists? Type below: _____________

Answer: In the given list the decimal is moving to the left by one place. From the scientific notation, numbers are decreasing by 10. The number missing is 7

Question 9. c. Make a conjecture about the missing numbers. Type below: _____________

Answer: The numbers will continue to decrease by 10 in the given list.

Conclusion:

Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation Free PDF for all the students. Students must go through the solved examples to have a complete grip on the maths and also on the way to solving each problem. Go Math Grade 8 Chapter 2 Exponents and Scientific Notation Answer Key will make students task easier to finish their maths practice. A good score will be in your hands by selecting the best way of learning. So, without any late begin your practice now.

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Exponents and Scientific Notation Activity Bundle 8th Grade

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This Exponents and Scientific Notation Activity Bundle 8th grade includes 6 classroom activities to support students’ knowledge of exponents and scientific notation for CCSS and TEKS.

These hands-on and engaging activities are all easy to prep! Students are able to practice and apply concepts with these exponents and scientific notation activities, while collaborating and having fun!  Math can be fun and interactive!

Standards: CCSS (8.EE.1, 8.EE.2, 8.EE.3, 8.EE.4) and TEKS (8.2C)

What is included in the 8th grade Exponents and Scientific Notation Activity Bundle?

Six hands-on activities that can be utilized in pairs or groups of 3-4. All activities include any necessary recording sheets and answer keys.

• Solve and Color:  properties of exponents
• Cut and Paste:  properties of exponents
• Puzzle:  square roots and cube roots
• Scavenger Hunt:  scientific notation
• Task Cards:  operations with scientific notation
• Stations:  exponents and scientific notation

How to use this resource:

• Use as a whole group classroom activity
• Use in a small group for additional remediation, tutoring, or enrichment
• Use as an alternative homework or independent practice assignment
• Incorporate within our CCSS- Aligned Exponents and Scientific Notation Unit  to support the mastery of concepts and skills.

Time to Complete:

• Most activities can be utilized within one class period. Performance tasks summarize the entire unit and may need 2-3 class periods. However, feel free to review the activities and select specific problems to meet your students’ needs and time specifications. There are multiple problems to practice the same concepts, so you can adjust as needed.

Looking for more 8 th Grade Math Material? Join our All Access Membership Community! You can reach your students without the “I still have to prep for tomorrow” stress, the constant overwhelm of teaching multiple preps, and the hamster wheel demands of creating your own teaching materials.

• Grade Level Curriculum
• Supplemental Digital Components
• Complete and Comprehensive Student Video Library 

Click here to learn more about All Access by Maneuvering the Middle®!

Licensing: This file is a license for ONE teacher and their students. Please purchase the appropriate number of licenses if you plan to use this resource with your team. Thank you!

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Maneuvering the Middle ® Terms of Use: Products by Maneuvering the Middle®, LLC may be used by the purchaser for their classroom use only. This is a single classroom license only. All rights reserved. Resources may only be posted online in an LMS such as Google Classroom, Canvas, or Schoology. Students should be the only ones able to access the resources.  It is a copyright violation to upload the files to school/district servers or shared Google Drives. See more information on our terms of use here . 

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This file is a license for one teacher and their students. Please purchase the appropriate number of licenses if you plan to use this resource with your team. Thank you!

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Digital Math Activity Bundle 8th Grade

Exponents and Scientific Notation Unit 8th Grade CCSS

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