unit exponents and scientific notation homework 6

Curriculum  /  Math  /  8th Grade  /  Unit 1: Exponents and Scientific Notation  /  Lesson 6

Exponents and Scientific Notation

Lesson 6 of 15

Criteria for Success

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

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

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

• Investigate, determine, and apply the general rule for power of product : $${(xy)^m = x^my^m}$$ .
• Investigate, determine, and apply the general rule for power of powers :  $${(x^m)^n=x^{mn}}$$ .
• Know that $${\left ( x+y \right )^{m} \neq x^m+y^m}$$ .

Suggestions for teachers to help them teach this lesson

• In terms of pacing, this lesson may be split over more than one day.
• Similar to Lesson 5, these Anchor Problems can be used in a variety of ways, including having students lead the discovery and seek out a general rule. 
• Once students have experimented with the problems and found a generalization, then provide them with the name of the rule and the general form.

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

Is the following statement true? Show your reasoning.

$${4^53^5=12^5}$$

Guiding Questions

Write an equivalent form of each of the following:

a.    $${(4x)^5}$$

b.    $${(-3mn)^2}$$

c.    $${\left ({5x\over y} \right )^3}$$

Lucas thinks that since $${(ab)^2 = a^2b^2}$$ , then that must mean $${(a+b)^2 = a^2+b^2}$$ . Is Lucas’ reasoning correct? Explain or show why or why not.

How is $${7^27^6}$$ different from $${(7^2)^6}$$ ? What is an equivalent expression for each one? 

Use your reasoning to simplify the following:

a.     $${(11^5)^4}$$

b.     $${-(2^3)^6}$$

c.     $${((-1)^3)^{12}}$$

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

Simplify the following expressions:

a.    $$(2^5)^7$$

b.    $$(91^3\times 19\times 103^8)^4$$

c.    $$(p^4q^5r)^9$$

d.    $$2^7\over 3^7$$

Student Response

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.

• Include a mixture of problems that involve using all the rules learned so far.
• Revisit the worksheet from Lesson 4, before students learned the more general approaches and rules. Do any of the problems illustrate the rules you’ve learned?
• EngageNY Mathematics Grade 8 Mathematics > Module 1 > Topic A > Lesson 3 — Exercises and Problem Set
• Kuta Software Free Pre-Algebra Worksheets Exponents and Radicals — Powers of products and quotients

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.

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

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

Simplify and write equivalent exponential expressions using all exponent rules.

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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5.2 Properties of Exponents and Scientific Notation

Learning objectives.

• Simplify expressions using the properties for exponents
• Use the definition of a negative exponent
• Use scientific notation

Be Prepared 5.2

Before you get started, take this readiness quiz.

• Simplify: ( −2 ) ( −2 ) ( −2 ) . ( −2 ) ( −2 ) ( −2 ) . If you missed this problem, review Example 1.19 .
• Simplify: 8 x 24 y . 8 x 24 y . If you missed this problem, review Example 1.24 .
• Name the decimal ( −2.6 ) ( 4.21 ) . ( −2.6 ) ( 4.21 ) . If you missed this problem, review Example 1.36 .

Simplify Expressions Using the Properties for Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, in the expression a m , a m , the exponent m tells us how many times we use the base a as a factor.

Let’s review the vocabulary for expressions with exponents.

Exponential Notation

This is read a to the m t h m t h power.

In the expression a m , a m , the exponent m tells us how many times we use the base a as a factor.

When we combine like terms by adding and subtracting, we need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

First, we will look at an example that leads to the Product Property .

Notice that 5 is the sum of the exponents, 2 and 3. We see x 2 · x 3 x 2 · x 3 is x 2 + 3 x 2 + 3 or x 5 . x 5 .

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Product Property for Exponents

If a is a real number and m and n are integers, then

To multiply with like bases, add the exponents.

Example 5.12

Simplify each expression: ⓐ y 5 · y 6 y 5 · y 6 ⓑ 2 x · 2 3 x 2 x · 2 3 x ⓒ 2 a 7 · 3 a . 2 a 7 · 3 a .

Try It 5.23

Simplify each expression:

ⓐ b 9 · b 8 b 9 · b 8 ⓑ 4 2 x · 4 x 4 2 x · 4 x ⓒ 3 p 5 · 4 p 3 p 5 · 4 p ⓓ x 6 · x 4 · x 8 . x 6 · x 4 · x 8 .

Try It 5.24

ⓐ x 12 · x 4 x 12 · x 4 ⓑ 10 · 10 x 10 · 10 x ⓒ 2 z · 6 z 7 2 z · 6 z 7 ⓓ b 5 · b 9 · b 5 . b 5 · b 9 · b 5 .

Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples.

Notice, in each case the bases were the same and we subtracted exponents. We see x 5 x 2 x 5 x 2 is x 5 − 2 x 5 − 2 or x 3 x 3 . We see x 2 x 3 x 2 x 3 is or 1 x . 1 x . When the larger exponent was in the numerator, we were left with factors in the numerator. When the larger exponent was in the denominator, we were left with factors in the denominator--notice the numerator of 1. When all the factors in the numerator have been removed, remember this is really dividing the factors to one, and so we need a 1 in the numerator. x x = 1 x x = 1 . This leads to the Quotient Property for Exponents.

Quotient Property for Exponents

If a is a real number, a ≠ 0 , a ≠ 0 , and m and n are integers, then

Example 5.13

Simplify each expression: ⓐ x 9 x 7 x 9 x 7 ⓑ 3 10 3 2 3 10 3 2 ⓒ b 8 b 12 b 8 b 12 ⓓ 7 3 7 5 . 7 3 7 5 .

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.

Try It 5.25

Simplify each expression: ⓐ x 15 x 10 x 15 x 10 ⓑ 6 14 6 5 6 14 6 5 ⓒ x 18 x 22 x 18 x 22 ⓓ 12 15 12 30 . 12 15 12 30 .

Try It 5.26

Simplify each expression: ⓐ y 43 y 37 y 43 y 37 ⓑ 10 15 10 7 10 15 10 7 ⓒ m 7 m 15 m 7 m 15 ⓓ 9 8 9 19 . 9 8 9 19 .

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like a m a m . a m a m . We know , x x = 1 , , x x = 1 , for any x ( x ≠ 0 ) x ( x ≠ 0 ) since any number divided by itself is 1.

The Quotient Property for Exponents shows us how to simplify a m a m . a m a m . when m > n m > n and when n < m n < m by subtracting exponents. What if m = n ? m = n ? We will simplify a m a m a m a m in two ways to lead us to the definition of the Zero Exponent Property . In general, for a ≠ 0 : a ≠ 0 :

We see a m a m a m a m simplifies to a 0 a 0 and to 1. So a 0 = 1 . a 0 = 1 . Any non-zero base raised to the power of zero equals 1.

Zero Exponent Property

If a is a non-zero number, then a 0 = 1 . a 0 = 1 .

If a is a non-zero number, then a to the power of zero equals 1.

Any non-zero number raised to the zero power is 1.

In this text, we assume any variable that we raise to the zero power is not zero.

Example 5.14

Simplify each expression: ⓐ 9 0 9 0 ⓑ n 0 . n 0 .

The definition says any non-zero number raised to the zero power is 1.

ⓐ 9 0 Use the definition of the zero exponent. 1 9 0 Use the definition of the zero exponent. 1

ⓑ n 0 Use the definition of the zero exponent. 1 n 0 Use the definition of the zero exponent. 1

To simplify the expression n raised to the zero power we just use the definition of the zero exponent. The result is 1.

Try It 5.27

Simplify each expression: ⓐ 11 0 11 0 ⓑ q 0 . q 0 .

Try It 5.28

Simplify each expression: ⓐ 23 0 23 0 ⓑ r 0 . r 0 .

Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents has two forms depending on whether the exponent is larger in the numerator or the denominator. What if we just subtract exponents regardless of which is larger?

Let’s consider x 2 x 5 . x 2 x 5 . We subtract the exponent in the denominator from the exponent in the numerator. We see x 2 x 5 x 2 x 5 is x 2 − 5 x 2 − 5 or x −3 . x −3 .

We can also simplify x 2 x 5 x 2 x 5 by dividing out common factors:

This implies that x −3 = 1 x 3 x −3 = 1 x 3 and it leads us to the definition of a negative exponent . If n is an integer and a ≠ 0 , a ≠ 0 , then a − n = 1 a n . a − n = 1 a n .

Let’s now look at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

1 a − n Use the definition of a negative exponent, a − n = 1 a n . 1 1 a n Simplify the complex fraction. 1 · a n 1 Multiply. a n 1 a − n Use the definition of a negative exponent, a − n = 1 a n . 1 1 a n Simplify the complex fraction. 1 · a n 1 Multiply. a n

This implies 1 a − n = a n 1 a − n = a n and is another form of the definition of Properties of Negative Exponents .

Properties of Negative Exponents

If n is an integer and a ≠ 0 , a ≠ 0 , then a − n = 1 a n a − n = 1 a n or 1 a − n = a n . 1 a − n = a n .

The negative exponent tells us we can rewrite 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 , x −3 , we will take one more step and write 1 x 3 . 1 x 3 . The answer is considered to be in simplest form when it has only positive exponents.

Example 5.15

Simplify each expression: ⓐ x −5 x −5 ⓑ 10 −3 10 −3 ⓒ 1 y −4 1 y −4 ⓓ 1 3 −2 . 1 3 −2 .

ⓐ x −5 Use the definition of a negative exponent, a − n = 1 a n . 1 x 5 x −5 Use the definition of a negative exponent, a − n = 1 a n . 1 x 5

ⓑ 10 −3 Use the definition of a negative exponent, a − n = 1 a n . 1 10 3 Simplify. 1 1000 10 −3 Use the definition of a negative exponent, a − n = 1 a n . 1 10 3 Simplify. 1 1000

ⓒ 1 y −4 Use the property of a negative exponent, 1 a − n = a n . y 4 1 y −4 Use the property of a negative exponent, 1 a − n = a n . y 4

ⓓ 1 3 −2 Use the property of a negative exponent, 1 a − n = a n . 3 2 Simplify. 9 1 3 −2 Use the property of a negative exponent, 1 a − n = a n . 3 2 Simplify. 9

Try It 5.29

Simplify each expression: ⓐ z −3 z −3 ⓑ 10 −7 10 −7 ⓒ 1 p −8 1 p −8 ⓓ 1 4 −3 . 1 4 −3 .

Try It 5.30

Simplify each expression: ⓐ n −2 n −2 ⓑ 10 −4 10 −4 ⓒ 1 q −7 1 q −7 ⓓ 1 2 −4 . 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.

( 3 4 ) −2 Use the definition of a negative exponent, a − n = 1 a n . 1 ( 3 4 ) 2 Simplify the denominator. 1 9 16 Simplify the complex fraction. 16 9 But we know that 16 9 is ( 4 3 ) 2 . This tells us that ( 3 4 ) −2 = ( 4 3 ) 2 ( 3 4 ) −2 Use the definition of a negative exponent, a − n = 1 a n . 1 ( 3 4 ) 2 Simplify the denominator. 1 9 16 Simplify the complex fraction. 16 9 But we know that 16 9 is ( 4 3 ) 2 . This tells us that ( 3 4 ) −2 = ( 4 3 ) 2

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 Power Property

If a and b are real numbers, a ≠ 0 , b ≠ 0 a ≠ 0 , b ≠ 0 and n is an integer, then

Example 5.16

Simplify each expression: ⓐ ( 5 7 ) −2 ( 5 7 ) −2 ⓑ ( − x y ) −3 . ( − x y ) −3 .

ⓐ ( 5 7 ) −2 Use the Quotient to a Negative Exponent Property, ( a b ) − n = ( b a ) n . Take the reciprocal of the fraction and change the sign of the exponent. ( 7 5 ) 2 Simplify. 49 25 ( 5 7 ) −2 Use the Quotient to a Negative Exponent Property, ( a b ) − n = ( b a ) n . Take the reciprocal of the fraction and change the sign of the exponent. ( 7 5 ) 2 Simplify. 49 25

ⓑ ( − x y ) −3 Use the Quotient to a Negative Exponent Property, ( a b ) − n = ( b a ) n . Take the reciprocal of the fraction and change the sign of the exponent. ( − y x ) 3 Simplify. − y 3 x 3 ( − x y ) −3 Use the Quotient to a Negative Exponent Property, ( a b ) − n = ( b a ) n . Take the reciprocal of the fraction and change the sign of the exponent. ( − y x ) 3 Simplify. − y 3 x 3

Try It 5.31

Simplify each expression: ⓐ ( 2 3 ) −4 ( 2 3 ) −4 ⓑ ( − m n ) −2 . ( − m n ) −2 .

Try It 5.32

Simplify each expression: ⓐ ( 3 5 ) −3 ( 3 5 ) −3 ⓑ ( − a b ) −4 . ( − a b ) −4 .

Now that we have negative exponents, we will use the Product Property with expressions that have negative exponents.

Example 5.17

Simplify each expression: ⓐ z −5 · z −3 z −5 · z −3 ⓑ ( m 4 n −3 ) ( m −5 n −2 ) ( m 4 n −3 ) ( m −5 n −2 ) ⓒ ( 2 x −6 y 8 ) ( −5 x 5 y −3 ) . ( 2 x −6 y 8 ) ( −5 x 5 y −3 ) .

ⓐ z −5 · z −3 Add the exponents, since the bases are the same. z −5 − 3 Simplify. z −8 Use the definition of a negative exponent. 1 z 8 z −5 · z −3 Add the exponents, since the bases are the same. z −5 − 3 Simplify. z −8 Use the definition of a negative exponent. 1 z 8

ⓑ ( m 4 n −3 ) ( m −5 n −2 ) Use the Commutative Property to get like bases together. m 4 m −5 · n −2 n −3 Add the exponents for each base. m −1 · n −5 Take reciprocals and change the signs of the exponents. 1 m 1 · 1 n 5 Simplify. 1 m n 5 ( m 4 n −3 ) ( m −5 n −2 ) Use the Commutative Property to get like bases together. m 4 m −5 · n −2 n −3 Add the exponents for each base. m −1 · n −5 Take reciprocals and change the signs of the exponents. 1 m 1 · 1 n 5 Simplify. 1 m n 5

ⓒ ( 2 x −6 y 8 ) ( −5 x 5 y −3 ) Rewrite with the like bases together. 2 ( −5 ) · ( x −6 x 5 ) · ( y 8 y −3 ) Multiply the coefficients and add the exponents of each variable. −10 · x −1 · y 5 Use the definition of a negative exponent, a − n = 1 a n . −10 · 1 x · y 5 Simplify. −10 y 5 x ( 2 x −6 y 8 ) ( −5 x 5 y −3 ) Rewrite with the like bases together. 2 ( −5 ) · ( x −6 x 5 ) · ( y 8 y −3 ) Multiply the coefficients and add the exponents of each variable. −10 · x −1 · y 5 Use the definition of a negative exponent, a − n = 1 a n . −10 · 1 x · y 5 Simplify. −10 y 5 x

Try It 5.33

ⓐ z −4 · z −5 z −4 · z −5 ⓑ ( p 6 q −2 ) ( p −9 q −1 ) ( p 6 q −2 ) ( p −9 q −1 ) ⓒ ( 3 u −5 v 7 ) ( −4 u 4 v −2 ) . ( 3 u −5 v 7 ) ( −4 u 4 v −2 ) .

Try It 5.34

ⓐ c −8 · c −7 c −8 · c −7 ⓑ ( r 5 s −3 ) ( r −7 s −5 ) ( r 5 s −3 ) ( r −7 s −5 ) ⓒ ( −6 c −6 d 4 ) ( −5 c −2 d −1 ) . ( −6 c −6 d 4 ) ( −5 c −2 d −1 ) .

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

( x 2 ) 3 What does this mean? x 2 · x 2 · x 2 ( x 2 ) 3 What does this mean? x 2 · x 2 · x 2

Notice the 6 is the product of the exponents, 2 and 3. We see that ( x 2 ) 3 ( x 2 ) 3 is x 2 · 3 x 2 · 3 or x 6 . x 6 .

We multiplied the exponents. This leads to the Power Property for Exponents.

Power Property for Exponents

To raise a power to a power, multiply the exponents.

Example 5.18

Simplify each expression: ⓐ ( y 5 ) 9 ( y 5 ) 9 ⓑ ( 4 4 ) 7 ( 4 4 ) 7 ⓒ ( y 3 ) 6 ( y 5 ) 4 . ( y 3 ) 6 ( y 5 ) 4 .

ⓒ ( y 3 ) 6 ( y 5 ) 4 Use the Power Property. y 18 · y 20 Add the exponents. y 38 ( y 3 ) 6 ( y 5 ) 4 Use the Power Property. y 18 · y 20 Add the exponents. y 38

Try It 5.35

Simplify each expression: ⓐ ( b 7 ) 5 ( b 7 ) 5 ⓑ ( 5 4 ) 3 ( 5 4 ) 3 ⓒ ( a 4 ) 5 ( a 7 ) 4 . ( a 4 ) 5 ( a 7 ) 4 .

Try It 5.36

Simplify each expression: ⓐ ( z 6 ) 9 ( z 6 ) 9 ⓑ ( 3 7 ) 7 ( 3 7 ) 7 ⓒ ( q 4 ) 5 ( q 3 ) 3 . ( q 4 ) 5 ( q 3 ) 3 .

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

( 2 x ) 3 What does this mean? 2 x · 2 x · 2 x We group the like factors together. 2 · 2 · 2 · x · x · x How many factors of 2 and of x 2 3 · x 3 ( 2 x ) 3 What does this mean? 2 x · 2 x · 2 x We group the like factors together. 2 · 2 · 2 · x · x · x How many factors of 2 and of x 2 3 · x 3

Notice that each factor was raised to the power and ( 2 x ) 3 ( 2 x ) 3 is 2 3 · x 3 . 2 3 · x 3 .

The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents .

Product to a Power Property for Exponents

If a and b are real numbers and m is a whole number, then

To raise a product to a power, raise each factor to that power.

Example 5.19

Simplify each expression: ⓐ ( −3 m n ) 3 ( −3 m n ) 3 ⓑ ( −4 a 2 b ) 0 ( −4 a 2 b ) 0 ⓒ ( 6 k 3 ) −2 ( 6 k 3 ) −2 ⓓ ( 5 x −3 ) 2 . ( 5 x −3 ) 2 .

ⓑ ( −4 a 2 b ) 0 Use Power of a Product Property, ( a b ) m = a m b m . ( −4 ) 0 ( a 2 ) 0 ( b ) 0 Simplify. 1 · 1 · 1 Multiply. 1 ( −4 a 2 b ) 0 Use Power of a Product Property, ( a b ) m = a m b m . ( −4 ) 0 ( a 2 ) 0 ( b ) 0 Simplify. 1 · 1 · 1 Multiply. 1

ⓒ ( 6 k 3 ) −2 Use the Product to a Power Property, ( a b ) m = a m b m . ( 6 ) −2 ( k 3 ) −2 Use the Power Property, ( a m ) n = a m · n . 6 −2 k −6 Use the Definition of a negative exponent, a − n = 1 a n . 1 6 2 · 1 k 6 Simplify. 1 36 k 6 ( 6 k 3 ) −2 Use the Product to a Power Property, ( a b ) m = a m b m . ( 6 ) −2 ( k 3 ) −2 Use the Power Property, ( a m ) n = a m · n . 6 −2 k −6 Use the Definition of a negative exponent, a − n = 1 a n . 1 6 2 · 1 k 6 Simplify. 1 36 k 6

ⓓ ( 5 x −3 ) 2 Use the Product to a Power Property, ( a b ) m = a m b m . 5 2 ( x −3 ) 2 Simplify. 25 · x −6 Rewrite x −6 using, a − n = 1 a n . 25 · 1 x 6 Simplify. 25 x 6 ( 5 x −3 ) 2 Use the Product to a Power Property, ( a b ) m = a m b m . 5 2 ( x −3 ) 2 Simplify. 25 · x −6 Rewrite x −6 using, a − n = 1 a n . 25 · 1 x 6 Simplify. 25 x 6

Try It 5.37

Simplify each expression: ⓐ ( 2 w x ) 5 ( 2 w x ) 5 ⓑ ( −11 p q 3 ) 0 ( −11 p q 3 ) 0 ⓒ ( 2 b 3 ) −4 ( 2 b 3 ) −4 ⓓ ( 8 a −4 ) 2 . ( 8 a −4 ) 2 .

Try It 5.38

Simplify each expression: ⓐ ( −3 y ) 3 ( −3 y ) 3 ⓑ ( −8 m 2 n 3 ) 0 ( −8 m 2 n 3 ) 0 ⓒ ( −4 x 4 ) −2 ( −4 x 4 ) −2 ⓓ ( 2 c −4 ) 3 . ( 2 c −4 ) 3 .

Now we will look at an example that will lead us to the Quotient to a Power Property.

( x y ) 3 This means x y · x y · x y Multiply the fractions. x · x · x y · y · y Write with exponents. x 3 y 3 ( x y ) 3 This means x y · x y · x y Multiply the fractions. x · x · x y · y · y Write with exponents. x 3 y 3

Notice that the exponent applies to both the numerator and the denominator.

We see that ( x y ) 3 ( x y ) 3 is x 3 y 3 . x 3 y 3 .

This leads to the Quotient to a Power Property for Exponents .

Quotient to a Power Property for Exponents

If a a and b b are real numbers, b ≠ 0 , b ≠ 0 , and m m is an integer, then

To raise a fraction to a power, raise the numerator and denominator to that power.

Example 5.20

ⓐ ( b 3 ) 4 ( b 3 ) 4 ⓑ ( k j ) −3 ( k j ) −3 ⓒ ( 2 x y 2 z ) 3 ( 2 x y 2 z ) 3 ⓓ ( 4 p −3 q 2 ) 2 . ( 4 p −3 q 2 ) 2 .

ⓒ ( 2 x y 2 z ) 3 Use Quotient to a Power Property, ( a b ) m = a m b m . ( 2 x y 2 ) 3 z 3 Use the Product to a Power Property, ( a b ) m = a m b m . 8 x 3 y 6 z 3 ( 2 x y 2 z ) 3 Use Quotient to a Power Property, ( a b ) m = a m b m . ( 2 x y 2 ) 3 z 3 Use the Product to a Power Property, ( a b ) m = a m b m . 8 x 3 y 6 z 3

ⓓ ( 4 p −3 q 2 ) 2 Use Quotient to a Power Property, ( a b ) m = a m b m . ( 4 p −3 ) 2 ( q 2 ) 2 Use the Product to a Power Property, ( a b ) m = a m b m . 4 2 ( p −3 ) 2 ( q 2 ) 2 Simplify using the Power Property, ( a m ) n = a m · n . 16 p −6 q 4 Use the definition of negative exponent. 16 q 4 · 1 p 6 Simplify. 16 p 6 q 4 ( 4 p −3 q 2 ) 2 Use Quotient to a Power Property, ( a b ) m = a m b m . ( 4 p −3 ) 2 ( q 2 ) 2 Use the Product to a Power Property, ( a b ) m = a m b m . 4 2 ( p −3 ) 2 ( q 2 ) 2 Simplify using the Power Property, ( a m ) n = a m · n . 16 p −6 q 4 Use the definition of negative exponent. 16 q 4 · 1 p 6 Simplify. 16 p 6 q 4

Try It 5.39

ⓐ ( p 10 ) 4 ( p 10 ) 4 ⓑ ( m n ) −7 ( m n ) −7 ⓒ ( 3 a b 3 c 2 ) 4 ( 3 a b 3 c 2 ) 4 ⓓ ( 3 x −2 y 3 ) 3 . ( 3 x −2 y 3 ) 3 .

Try It 5.40

ⓐ ( −2 q ) 3 ( −2 q ) 3 ⓑ ( w x ) −4 ( w x ) −4 ⓒ ( x y 3 3 z 2 ) 2 ( x y 3 3 z 2 ) 2 ⓓ ( 2 m −2 n −2 ) 3 . ( 2 m −2 n −2 ) 3 .

We now have several properties for exponents. Let’s summarize them and then we’ll do some more examples that use more than one of the properties.

Summary of Exponent Properties

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

Example 5.21

Simplify each expression by applying several properties:

ⓐ ( 3 x 2 y ) 4 ( 2 x y 2 ) 3 ( 3 x 2 y ) 4 ( 2 x y 2 ) 3 ⓑ ( x 3 ) 4 ( x −2 ) 5 ( x 6 ) 5 ( x 3 ) 4 ( x −2 ) 5 ( x 6 ) 5 ⓒ ( 2 x y 2 x 3 y −2 ) 2 ( 12 x y 3 x 3 y −1 ) −1 . ( 2 x y 2 x 3 y −2 ) 2 ( 12 x y 3 x 3 y −1 ) −1 .

ⓐ ( 3 x 2 y ) 4 ( 2 x y 2 ) 3 Use the Product to a Power Property, ( a b ) m = a m b m . ( 3 4 x 8 y 4 ) ( 2 3 x 3 y 6 ) Simplify. ( 81 x 8 y 4 ) ( 8 x 3 y 6 ) Use the Commutative Property. 81 · 8 · x 8 · x 3 · y 4 · y 6 Multiply the constants and add the exponents. 648 x 11 y 10 ( 3 x 2 y ) 4 ( 2 x y 2 ) 3 Use the Product to a Power Property, ( a b ) m = a m b m . ( 3 4 x 8 y 4 ) ( 2 3 x 3 y 6 ) Simplify. ( 81 x 8 y 4 ) ( 8 x 3 y 6 ) Use the Commutative Property. 81 · 8 · x 8 · x 3 · y 4 · y 6 Multiply the constants and add the exponents. 648 x 11 y 10

ⓑ ( x 3 ) 4 ( x −2 ) 5 ( x 6 ) 5 Use the Power Property, ( a m ) n = a m · n . ( x 12 ) ( x −10 ) ( x 30 ) Add the exponents in the numerator. x 2 x 30 Use the Quotient Property, a m a n = 1 a n − m . 1 x 28 ( x 3 ) 4 ( x −2 ) 5 ( x 6 ) 5 Use the Power Property, ( a m ) n = a m · n . ( x 12 ) ( x −10 ) ( x 30 ) Add the exponents in the numerator. x 2 x 30 Use the Quotient Property, a m a n = 1 a n − m . 1 x 28

ⓒ ( 2 x y 2 x 3 y −2 ) 2 ( 12 x y 3 x 3 y −1 ) −1 Simplify inside the parentheses first. ( 2 y 4 x 2 ) 2 ( 12 y 4 x 2 ) −1 Use the Quotient to a Power Property, ( a b ) m = a m b m . ( 2 y 4 ) 2 ( x 2 ) 2 ( 12 y 4 ) −1 ( x 2 ) −1 Use the Product to a Power Property, ( a b ) m = a m b m . 4 y 8 x 4 · 12 −1 y −4 x −2 Simplify. 4 y 4 12 x 2 Simplify. y 4 3 x 2 ( 2 x y 2 x 3 y −2 ) 2 ( 12 x y 3 x 3 y −1 ) −1 Simplify inside the parentheses first. ( 2 y 4 x 2 ) 2 ( 12 y 4 x 2 ) −1 Use the Quotient to a Power Property, ( a b ) m = a m b m . ( 2 y 4 ) 2 ( x 2 ) 2 ( 12 y 4 ) −1 ( x 2 ) −1 Use the Product to a Power Property, ( a b ) m = a m b m . 4 y 8 x 4 · 12 −1 y −4 x −2 Simplify. 4 y 4 12 x 2 Simplify. y 4 3 x 2

Try It 5.41

ⓐ ( c 4 d 2 ) 5 ( 3 c d 5 ) 4 ( c 4 d 2 ) 5 ( 3 c d 5 ) 4 ⓑ ( a −2 ) 3 ( a 2 ) 4 ( a 4 ) 5 ( a −2 ) 3 ( a 2 ) 4 ( a 4 ) 5 ⓒ ( 3 x y 2 x 2 y −3 ) 2 ( 9 x y −3 x 3 y 2 ) −1 . ( 3 x y 2 x 2 y −3 ) 2 ( 9 x y −3 x 3 y 2 ) −1 .

Try It 5.42

ⓐ ( a 3 b 2 ) 6 ( 4 a b 3 ) 4 ( a 3 b 2 ) 6 ( 4 a b 3 ) 4 ⓑ ( p −3 ) 4 ( p 5 ) 3 ( p 7 ) 6 ( p −3 ) 4 ( p 5 ) 3 ( p 7 ) 6 ⓒ ( 4 x 3 y 2 x 2 y −1 ) 2 ( 8 x y −3 x 2 y ) −1 . ( 4 x 3 y 2 x 2 y −1 ) 2 ( 8 x y −3 x 2 y ) −1 .

Use Scientific Notation

Working with very large or very small numbers can be awkward. Since our number system is base ten we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. Consider the numbers 4,000 and 0.004.

Using place value, we can rewrite the numbers 4,000 and 0.004. We know that 4,000 means 4 × 1,000 4 × 1,000 and 0.004 means 4 × 1 1,000 . 4 × 1 1,000 .

If we write the 1,000 as a power of ten in exponential form, we can rewrite these numbers in this way:

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 ten, 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

It is customary in scientific notation to use as the × × 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.

The power of 10 is positive when the number is larger than 1: 4,000 = 4 × 10 3 4,000 = 4 × 10 3

The power of 10 is negative when the number is between 0 and 1: 0.004 = 4 × 10 −3 0.004 = 4 × 10 −3

To convert a decimal 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 . 10 n .
• between 0 and 1, the power of 10 will be 10 − n . 10 − n .
• Step 4. Check.

Example 5.22

Write in scientific notation: ⓐ 37,000 ⓑ 0.0052 . 0.0052 .

Try It 5.43

Write in scientific notation: ⓐ 96,000 ⓑ 0.0078.

Try It 5.44

Write in scientific notation: ⓐ 48,300 ⓑ 0.0129.

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

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.

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.

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 | | n | places to the left.
• Step 3. Check.

Example 5.23

Convert to decimal form: ⓐ 6.2 × 10 3 6.2 × 10 3 ⓑ −8.9 × 10 −2 . −8.9 × 10 −2 .

Try It 5.45

Convert to decimal form: ⓐ 1.3 × 10 3 1.3 × 10 3 ⓑ −1.2 × 10 −4 . −1.2 × 10 −4 .

Try It 5.46

Convert to decimal form: ⓐ −9.5 × 10 4 −9.5 × 10 4 ⓑ 7.5 × 10 −2 . 7.5 × 10 −2 .

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 5.24

Multiply or divide as indicated. Write answers in decimal form: ⓐ ( −4 × 10 5 ) ( 2 × 10 −7 ) ( −4 × 10 5 ) ( 2 × 10 −7 ) ⓑ 9 × 10 3 3 × 10 −2 . 9 × 10 3 3 × 10 −2 .

ⓐ ( −4 × 10 5 ) ( 2 × 10 −7 ) Use the Commutative Property to rearrange the factors. −4 · 2 · 10 5 · 10 −7 Multiply. −8 × 10 −2 Change to decimal form by moving the decimal two places left. −0.08 ( −4 × 10 5 ) ( 2 × 10 −7 ) Use the Commutative Property to rearrange the factors. −4 · 2 · 10 5 · 10 −7 Multiply. −8 × 10 −2 Change to decimal form by moving the decimal two places left. −0.08

ⓑ 9 × 10 3 9 × 10 −2 Separate the factors, rewriting as the product of two fractions. 9 3 × 10 3 10 −2 Divide. 3 × 10 5 Change to decimal form by moving the decimal five places right. 300,000 9 × 10 3 9 × 10 −2 Separate the factors, rewriting as the product of two fractions. 9 3 × 10 3 10 −2 Divide. 3 × 10 5 Change to decimal form by moving the decimal five places right. 300,000

Try It 5.47

Multiply or divide as indicated. Write answers in decimal form:

ⓐ ( −3 × 10 5 ) ( 2 × 10 −8 ) ( −3 × 10 5 ) ( 2 × 10 −8 ) ⓑ 8 × 10 2 4 × 10 −2 . 8 × 10 2 4 × 10 −2 .

Try It 5.48

ⓐ ; ( −3 × 10 −2 ) ( 3 × 10 −1 ) ( −3 × 10 −2 ) ( 3 × 10 −1 ) ⓑ 8 × 10 4 2 × 10 −1 . 8 × 10 4 2 × 10 −1 .

Access these online resources for additional instruction and practice with using multiplication properties of exponents.

• Properties of Exponents
• Negative exponents

Section 5.2 Exercises

Practice makes perfect.

In the following exercises, simplify each expression using the properties for exponents.

ⓐ d 3 · d 6 d 3 · d 6 ⓑ 4 5 x · 4 9 x 4 5 x · 4 9 x ⓒ 2 y · 4 y 3 2 y · 4 y 3 ⓓ w · w 2 · w 3 w · w 2 · w 3

ⓐ x 4 · x 2 x 4 · x 2 ⓑ 8 9 x · 8 3 8 9 x · 8 3 ⓒ 3 z 25 · 5 z 8 3 z 25 · 5 z 8 ⓓ y · y 3 · y 5 y · y 3 · y 5

ⓐ n 19 · n 12 n 19 · n 12 ⓑ 3 x · 3 6 3 x · 3 6 ⓒ 7 w 5 · 8 w 7 w 5 · 8 w ⓓ a 4 · a 3 · a 9 a 4 · a 3 · a 9

ⓐ q 27 · q 15 q 27 · q 15 ⓑ 5 x · 5 4 x 5 x · 5 4 x ⓒ 9 u 41 · 7 u 53 9 u 41 · 7 u 53 ⓓ c 5 · c 11 · c 2 c 5 · c 11 · c 2

m x · m 3 m x · m 3

n y · n 2 n y · n 2

y a · y b y a · y b

x p · x q x p · x q

ⓐ x 18 x 3 x 18 x 3 ⓑ 5 12 5 3 5 12 5 3 ⓒ q 18 q 36 q 18 q 36 ⓓ 10 2 10 3 10 2 10 3

ⓐ y 20 y 10 y 20 y 10 ⓑ 7 16 7 2 7 16 7 2 ⓒ t 10 t 40 t 10 t 40 ⓓ 8 3 8 5 8 3 8 5

ⓐ p 21 p 7 p 21 p 7 ⓑ 4 16 4 4 4 16 4 4 ⓒ b b 9 b b 9 ⓓ 4 4 6 4 4 6

ⓐ u 24 u 3 u 24 u 3 ⓑ 9 15 9 5 9 15 9 5 ⓒ x x 7 x x 7 ⓓ 10 10 3 10 10 3

ⓐ 20 0 20 0 ⓑ b 0 b 0

ⓐ 13 0 13 0 ⓑ k 0 k 0

ⓐ − 27 0 − 27 0 ⓑ − ( 27 0 ) − ( 27 0 )

ⓐ − 15 0 − 15 0 ⓑ − ( 15 0 ) − ( 15 0 )

In the following exercises, simplify each expression.

ⓐ a −2 a −2 ⓑ 10 −3 10 −3 ⓒ 1 c −5 1 c −5 ⓓ 1 3 −2 1 3 −2

ⓐ b −4 b −4 ⓑ 10 −2 10 −2 ⓒ 1 c −5 1 c −5 ⓓ 1 5 −2 1 5 −2

ⓐ r −3 r −3 ⓑ 10 −5 10 −5 ⓒ 1 q −10 1 q −10 ⓓ 1 10 −3 1 10 −3

ⓐ s −8 s −8 ⓑ 10 −2 10 −2 ⓒ 1 t −9 1 t −9 ⓓ 1 10 −4 1 10 −4

ⓐ ( 5 8 ) −2 ( 5 8 ) −2 ⓑ ( − b a ) −2 ( − b a ) −2

ⓐ ( 3 10 ) −2 ( 3 10 ) −2 ⓑ ( − 2 z ) −3 ( − 2 z ) −3

ⓐ ( 4 9 ) −3 ( 4 9 ) −3 ⓑ ( − u v ) −5 ( − u v ) −5

ⓐ ( 7 2 ) −3 ( 7 2 ) −3 ⓑ ( − 3 x ) −3 ( − 3 x ) −3

ⓐ ( −5 ) −2 ( −5 ) −2 ⓑ − 5 −2 − 5 −2 ⓒ ( − 1 5 ) −2 ( − 1 5 ) −2 ⓓ − ( 1 5 ) −2 − ( 1 5 ) −2

ⓐ − 5 −3 − 5 −3 ⓑ ( − 1 5 ) −3 ( − 1 5 ) −3 ⓒ − ( 1 5 ) −3 − ( 1 5 ) −3 ⓓ ( −5 ) −3 ( −5 ) −3

ⓐ 3 · 5 −1 3 · 5 −1 ⓑ ( 3 · 5 ) −1 ( 3 · 5 ) −1

ⓐ 3 · 4 −2 3 · 4 −2 ⓑ ( 3 · 4 ) −2 ( 3 · 4 ) −2

In the following exercises, simplify each expression using the Product Property.

ⓐ b 4 b −8 b 4 b −8 ⓑ ( w 4 x −5 ) ( w −2 x −4 ) ( w 4 x −5 ) ( w −2 x −4 ) ⓒ ( −6 c −3 d 9 ) ( 2 c 4 d −5 ) ( −6 c −3 d 9 ) ( 2 c 4 d −5 )

ⓐ s 3 · s −7 s 3 · s −7 ⓑ ( m 3 n −3 ) ( m −5 n −1 ) ( m 3 n −3 ) ( m −5 n −1 ) ⓒ ( −2 j −5 k 8 ) ( 7 j 2 k −3 ) ( −2 j −5 k 8 ) ( 7 j 2 k −3 )

ⓐ a 3 · a −3 a 3 · a −3 ⓑ ( u v −2 ) ( u −5 v −3 ) ( u v −2 ) ( u −5 v −3 ) ⓒ ( −4 r −2 s −8 ) ( 9 r 4 s 3 ) ( −4 r −2 s −8 ) ( 9 r 4 s 3 )

ⓐ y 5 · y −5 y 5 · y −5 ⓑ ( p q −4 ) ( p −6 q −3 ) ( p q −4 ) ( p −6 q −3 ) ⓒ ( −5 m 4 n 6 ) ( 8 m −5 n −3 ) ( −5 m 4 n 6 ) ( 8 m −5 n −3 )

p 5 · p −2 · p −4 p 5 · p −2 · p −4

x 4 · x −2 · x −3 x 4 · x −2 · x −3

In the following exercises, simplify each expression using the Power Property.

ⓐ ( m 4 ) 2 ( m 4 ) 2 ⓑ ( 10 3 ) 6 ( 10 3 ) 6 ⓒ ( x 3 ) −4 ( x 3 ) −4

ⓐ ( b 2 ) 7 ( b 2 ) 7 ⓑ ( 3 8 ) 2 ( 3 8 ) 2 ⓒ ( k 2 ) −5 ( k 2 ) −5

ⓐ ( y 3 ) x ( y 3 ) x ⓑ ( 5 x ) y ( 5 x ) y ⓒ ( q 6 ) −8 ( q 6 ) −8

ⓐ ( x 2 ) y ( x 2 ) y ⓑ ( 7 a ) b ( 7 a ) b ⓒ ( a 9 ) −10 ( a 9 ) −10

In the following exercises, simplify each expression using the Product to a Power Property.

ⓐ ( −3 x y ) 2 ( −3 x y ) 2 ⓑ ( 6 a ) 0 ( 6 a ) 0 ⓒ ( 5 x 2 ) −2 ( 5 x 2 ) −2 ⓓ ( −4 y −3 ) 2 ( −4 y −3 ) 2

ⓐ ( −4 a b ) 2 ( −4 a b ) 2 ⓑ ( 5 x ) 0 ( 5 x ) 0 ⓒ ( 4 y 3 ) −3 ( 4 y 3 ) −3 ⓓ ( −7 y −3 ) 2 ( −7 y −3 ) 2

ⓐ ( −5 a b ) 3 ( −5 a b ) 3 ⓑ ( −4 p q ) 0 ( −4 p q ) 0 ⓒ ( −6 x 3 ) −2 ( −6 x 3 ) −2 ⓓ ( 3 y −4 ) 2 ( 3 y −4 ) 2

ⓐ ( −3 x y z ) 4 ( −3 x y z ) 4 ⓑ ( −7 m n ) 0 ( −7 m n ) 0 ⓒ ( −3 x 3 ) −2 ( −3 x 3 ) −2 ⓓ ( 2 y −5 ) 2 ( 2 y −5 ) 2

In the following exercises, simplify each expression using the Quotient to a Power Property.

ⓐ ( p 2 ) 5 ( p 2 ) 5 ⓑ ( x y ) −6 ( x y ) −6 ⓒ ( 2 x y 2 z ) 3 ( 2 x y 2 z ) 3 ⓓ ( 4 p −3 q 2 ) 2 ( 4 p −3 q 2 ) 2

ⓐ ( x 3 ) 4 ( x 3 ) 4 ⓑ ( a b ) −5 ( a b ) −5 ⓒ ( 2 x y 2 z ) 3 ( 2 x y 2 z ) 3 ⓓ ( x 3 y z 4 ) 2 ( x 3 y z 4 ) 2

ⓐ ( a 3 b ) 4 ( a 3 b ) 4 ⓑ ( 5 4 m ) −2 ( 5 4 m ) −2 ⓒ ( 3 a -2 b 3 c 3 ) -2 ( 3 a -2 b 3 c 3 ) -2 ⓓ ( p -1 q 4 r -4 ) 2 ( p -1 q 4 r -4 ) 2

ⓐ ( x 2 y ) 3 ( x 2 y ) 3 ⓑ ( 10 3 q ) −4 ( 10 3 q ) −4 ⓒ ( 2 x 3 y 4 3 z 2 ) 5 ( 2 x 3 y 4 3 z 2 ) 5 ⓓ ( 5 a 3 b -1 2 c 4 ) -3 ( 5 a 3 b -1 2 c 4 ) -3

In the following exercises, simplify each expression by applying several properties.

ⓐ ( 5 t 2 ) 3 ( 3 t ) 2 ( 5 t 2 ) 3 ( 3 t ) 2 ⓑ ( t 2 ) 5 ( t −4 ) 2 ( t 3 ) 7 ( t 2 ) 5 ( t −4 ) 2 ( t 3 ) 7 ⓒ ( 2 x y 2 x 3 y −2 ) 2 ( 12 x y 3 x 3 y −1 ) −1 ( 2 x y 2 x 3 y −2 ) 2 ( 12 x y 3 x 3 y −1 ) −1

ⓐ ( 10 k 4 ) 3 ( 5 k 6 ) 2 ( 10 k 4 ) 3 ( 5 k 6 ) 2 ⓑ ( q 3 ) 6 ( q −2 ) 3 ( q 4 ) 8 ( q 3 ) 6 ( q −2 ) 3 ( q 4 ) 8

ⓐ ( m 2 n ) 2 ( 2 m n 5 ) 4 ( m 2 n ) 2 ( 2 m n 5 ) 4 ⓑ ( −2 p −2 ) 4 ( 3 p 4 ) 2 ( −6 p 3 ) 2 ( −2 p −2 ) 4 ( 3 p 4 ) 2 ( −6 p 3 ) 2

ⓐ ( 3 p q 4 ) 2 ( 6 p 6 q ) 2 ( 3 p q 4 ) 2 ( 6 p 6 q ) 2 ⓑ ( −2 k −3 ) 2 ( 6 k 2 ) 4 ( 9 k 4 ) 2 ( −2 k −3 ) 2 ( 6 k 2 ) 4 ( 9 k 4 ) 2

Mixed Practice

ⓐ 7 n −1 7 n −1 ⓑ ( 7 n ) −1 ( 7 n ) −1 ⓒ ( −7 n ) −1 ( −7 n ) −1

ⓐ 6 r −1 6 r −1 ⓑ ( 6 r ) −1 ( 6 r ) −1 ⓒ ( −6 r ) −1 ( −6 r ) −1

ⓐ ( 3 p ) −2 ( 3 p ) −2 ⓑ 3 p −2 3 p −2 ⓒ −3 p −2 −3 p −2

ⓐ ( 2 q ) −4 ( 2 q ) −4 ⓑ 2 q −4 2 q −4 ⓒ −2 q −4 −2 q −4

( x 2 ) 4 · ( x 3 ) 2 ( x 2 ) 4 · ( x 3 ) 2

( y 4 ) 3 · ( y 5 ) 2 ( y 4 ) 3 · ( y 5 ) 2

( a 2 ) 6 · ( a 3 ) 8 ( a 2 ) 6 · ( a 3 ) 8

( b 7 ) 5 · ( b 2 ) 6 ( b 7 ) 5 · ( b 2 ) 6

( 2 m 6 ) 3 ( 2 m 6 ) 3

( 3 y 2 ) 4 ( 3 y 2 ) 4

( 10 x 2 y ) 3 ( 10 x 2 y ) 3

( 2 m n 4 ) 5 ( 2 m n 4 ) 5

( −2 a 3 b 2 ) 4 ( −2 a 3 b 2 ) 4

( −10 u 2 v 4 ) 3 ( −10 u 2 v 4 ) 3

( 2 3 x 2 y ) 3 ( 2 3 x 2 y ) 3

( 7 9 p q 4 ) 2 ( 7 9 p q 4 ) 2

( 8 a 3 ) 2 ( 2 a ) 4 ( 8 a 3 ) 2 ( 2 a ) 4

( 5 r 2 ) 3 ( 3 r ) 2 ( 5 r 2 ) 3 ( 3 r ) 2

( 10 p 4 ) 3 ( 5 p 6 ) 2 ( 10 p 4 ) 3 ( 5 p 6 ) 2

( 4 x 3 ) 3 ( 2 x 5 ) 4 ( 4 x 3 ) 3 ( 2 x 5 ) 4

( 1 2 x 2 y 3 ) 4 ( 4 x 5 y 3 ) 2 ( 1 2 x 2 y 3 ) 4 ( 4 x 5 y 3 ) 2

( 1 3 m 3 n 2 ) 4 ( 9 m 8 n 3 ) 2 ( 1 3 m 3 n 2 ) 4 ( 9 m 8 n 3 ) 2

( 3 m 2 n ) 2 ( 2 m n 5 ) 4 ( 3 m 2 n ) 2 ( 2 m n 5 ) 4

( 2 p q 4 ) 3 ( 5 p 6 q ) 2 ( 2 p q 4 ) 3 ( 5 p 6 q ) 2

ⓐ ( 3 x ) 2 ( 5 x ) ( 3 x ) 2 ( 5 x ) ⓑ ( 2 y ) 3 ( 6 y ) ( 2 y ) 3 ( 6 y )

ⓐ ( 1 2 y 2 ) 3 ( 2 3 y ) 2 ( 1 2 y 2 ) 3 ( 2 3 y ) 2 ⓑ ( 1 2 j 2 ) 5 ( 2 5 j 3 ) 2 ( 1 2 j 2 ) 5 ( 2 5 j 3 ) 2

ⓐ ( 2 r −2 ) 3 ( 4 −1 r ) 2 ( 2 r −2 ) 3 ( 4 −1 r ) 2 ⓑ ( 3 x −3 ) 3 ( 3 −1 x 5 ) 4 ( 3 x −3 ) 3 ( 3 −1 x 5 ) 4

( k −2 k 8 k 3 ) 2 ( k −2 k 8 k 3 ) 2

( j −2 j 5 j 4 ) 3 ( j −2 j 5 j 4 ) 3

( −4 m −3 ) 2 ( 5 m 4 ) 3 ( −10 m 6 ) 3 ( −4 m −3 ) 2 ( 5 m 4 ) 3 ( −10 m 6 ) 3

( −10 n −2 ) 3 ( 4 n 5 ) 2 ( 2 n 8 ) 2 ( −10 n −2 ) 3 ( 4 n 5 ) 2 ( 2 n 8 ) 2

In the following exercises, write each number in scientific notation.

ⓐ 57,000 ⓑ 0.026

ⓐ 340,000 ⓑ 0.041

ⓐ 8,750,000 ⓑ 0.00000871

ⓐ 1,290,000 ⓑ 0.00000103

In the following exercises, convert each number to decimal form.

ⓐ 5.2 × 10 2 5.2 × 10 2 ⓑ 2.5 × 10 −2 2.5 × 10 −2

ⓐ −8.3 × 10 2 −8.3 × 10 2 ⓑ 3.8 × 10 −2 3.8 × 10 −2

ⓐ 7.5 × 10 6 7.5 × 10 6 ⓑ −4.13 × 10 −5 −4.13 × 10 −5

ⓐ 1.6 × 10 10 1.6 × 10 10 ⓑ 8.43 × 10 −6 8.43 × 10 −6

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

ⓐ ( 3 × 10 −5 ) ( 3 × 10 9 ) ( 3 × 10 −5 ) ( 3 × 10 9 ) ⓑ 7 × 10 −3 1 × 10 −7 7 × 10 −3 1 × 10 −7

ⓐ ( 2 × 10 2 ) ( 1 × 10 −4 ) ( 2 × 10 2 ) ( 1 × 10 −4 ) ⓑ 5 × 10 −2 1 × 10 −10 5 × 10 −2 1 × 10 −10

ⓐ ( 7.1 × 10 −2 ) ( 2.4 × 10 −4 ) ( 7.1 × 10 −2 ) ( 2.4 × 10 −4 ) ⓑ 6 × 10 4 3 × 10 −2 6 × 10 4 3 × 10 −2

ⓐ ( 3.5 × 10 −4 ) ( 1.6 × 10 −2 ) ( 3.5 × 10 −4 ) ( 1.6 × 10 −2 ) ⓑ 8 × 10 6 4 × 10 −1 8 × 10 6 4 × 10 −1

Writing Exercises

Use the Product Property for Exponents to explain why x · x = x 2 . x · x = x 2 .

Jennifer thinks the quotient a 24 a 6 a 24 a 6 simplifies to a 4 . a 4 . What is wrong with her reasoning?

Explain why − 5 3 = ( −5 ) 3 − 5 3 = ( −5 ) 3 but − 5 4 ≠ ( −5 ) 4 . − 5 4 ≠ ( −5 ) 4 .

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all goals?

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Unit 7 - Exponents and Scientific Notation

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Maneuvering the Middle

Student-Centered Math Lessons

Teaching Scientific Notation and Exponents

Let’s chat about scientific notation and exponents !  I have found that the simplest skills in math are often the most miscalculated and confusing for students. Exponents and scientific notation can fall into this trap. 

Vertical Alignment

Exponent Tips

It is important with both exponents and scientific notation that students understand that they show a different way to represent a value.

Before even showing an exponent, start by showing the expanded form like  7*7*7*7*7. You can start by asking students:

• Is this the most efficient way to write this?
• Would 7*5 give you the same result? Why or why not?

Tip: I have learned the hard way to NEVER use 2^2 in any of your early examples because it will just confuse students into thinking you multiply the base and exponent.

Easy, no-plan activity idea: A fun way for students to practice exponents is by using concentric circles. The inside circle is the base and the outside circle is the exponent. Assign students numbers 1-10 and have them rotate to a new partner each round. Students pair up, write down the exponent form, the expanded form, and then calculate the standard form. Keep rotating until your time is up. 

Laws of Exponents

The laws of exponents are so fun! I love how students can build on their previous knowledge to come up with the laws themselves. For example: 

On a Facebook thread, I recently saw a teacher say, “When in doubt, expand it out.” If a student forgets a law, all they need to do is expand it and calculate to discover the law again. That is something a student is more likely to do if you are modeling it consistently.

Because the laws are so accessible, this content really shines as a discovery-based lesson. Your students could also participate in a jigsaw. Each group becomes experts at their assigned law, then they present the law, the proof, and examples to their peers. 

If you go the traditional teaching route, I recommend splitting this skill up over at least 2 days. Maneuvering the Middle 8th grade curriculum covers multiplying/dividing like bases, power to power, and product to power on day one. Negative and zero exponents are covered on day two.

I highly recommend an anchor chart with all of the laws for easy reference. Sometimes in my last class on a Friday, my brain needed to look at an anchor chart to give it the boost it needed (and I am the teacher).

Scientific Notation

Like I said before, scientific notation is just a different way to represent a value. Here is a great way to introduce why we might use scientific notation. Write down the mass of Earth and Mars on your whiteboard or project it. Make sure students will have to copy it down themselves since that is part of your point. 

Start by asking students to read the numbers to you. You will get some funny responses. Then ask students to add them or subtract the masses. As students write and count all of the zeros and inevitably miscalculate or miscount the number of zeros, you can introduce why we used scientific notation. (Less room for error, more efficient) Scientific notation is similar to typing TTYL instead of typing “talk to you later.”

Tips for Scientific Notation

Avoid using right or left when describing the direction to move the decimal. Instead, emphasize that smaller numbers will have negative exponents and larger numbers will have positive exponents. This re-enforces the negative exponent law.

Speaking of exponent laws, scientific notation operations reinforce the laws of exponents. If you look at the vertical alignment, scientific notation only shows up in 8th grade (in TEKS and CCSS), so at least, it reinforces other important concepts that students will use in Algebra 1 and 2.

I have never (and will never) teach Science, but it did occur to me to look up the Texas Science standards, and take a look at this chemistry standard – 

“C.2(G)  express and manipulate chemical quantities using scientific conventions and mathematical procedures, including dimensional analysis, scientific notation , and significant figures”

An opportunity for cross curricular?! Wahoo! If this is something that has peaked your interest, here is a NASA themed exploration lesson with resources for practicing scientific notation. This demos activity is also a great science based activity.

What tips do you have for teaching exponents and scientific notation?

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Chapter 6: Polynomials

6.1 Working with Exponents

Exponents often can be simplified using a few basic properties, since exponents represent repeated multiplication. The basic structure of writing an exponent looks like [latex]x^y,[/latex] where [latex]x[/latex] is defined as the base and [latex]y[/latex] is termed its exponent. For this instance, [latex]y[/latex] represents the number of times that the variable [latex]x[/latex] is multiplied by itself

When looking at numbers to various powers, the following table gives the numeric value of several numbers to various powers.

[latex]\begin{array}{llllll} \text{Squares}&\text{Cubes}&4^{\text{th}}\text{ Power}&5^{\text{th}}\text{ Power}&6^{\text{th}}\text{ Power}&7^{\text{th}}\text{ Power} \\ \\ 2^2=4&2^3=8&2^4=16&2^5=32&2^6=64&2^7=128 \\ 3^2=9&3^3=27&3^4=81&3^5=243&3^6=729&3^7=2,187 \\ 4^2=16&4^3=64&4^4=256&4^5=1,024&4^6=4,096&4^7=16,384 \\ 5^2=25&5^3=125&5^4=625&5^5=3,125&5^6=15,625&5^7=78,125 \\ 6^2=36&6^3=216&6^4=1,296&6^5=7,776&6^6=46,656&6^7=279,936 \\ 7^2=49&7^3=343&7^4=2,401&7^5=16,807&7^6=117,649&7^7=823,543 \\ 8^2=64&8^3=512&8^4=4,096&8^5=32,768&8^6=262,144&8^7=2,097,152 \\ 9^2=81&9^3=729&9^4=6,561&9^5=59,049&9^6=531,441&9^7=4,782,969 \\ 10^2=100&10^3=1,000&10^4=10,000&10^5=100,000&10^6=1,000,000&10^7=10,000,000 \\ \\ 11^2=121&12^2=144&13^2=169&14^2=196&15^2=225&20^2=400 \end{array}[/latex]

For this chart, the expanded forms of the base 2 for multiple exponents is shown:

[latex]\begin{array}{lllllllllllllll} 2^2&=&2&\times &2&=&4,&&&&&&&& \\ 2^3&=&2&\times &2&\times &2&=&8,&&&&&& \\ 2^4&=&2&\times &2&\times &2&\times &2&=&16,&&&& \\ 2^5&=&2&\times &2&\times &2&\times &2&\times &2&=&32&& \\ 2^6&=&2&\times &2&\times &2&\times &2&\times &2&\times &2&=&64\hspace{0.25in} \text{and so on} \\ \end{array}[/latex]

Once there is an exponent as a base that is multiplied or divided by itself to the number represented by the exponent, it becomes straightforward to identify a number of rules and properties that can be defined.

The following examples outline a number of these rules.

Example 6.1.1

What is the value of [latex]a^2 \times a^3[/latex]?

[latex]a^2 \times a^3[/latex] means that you have [latex](a \times a) (a \times a \times a),[/latex]

which is the same as [latex](a \times a \times a \times a \times a)[/latex]

or [latex]a^5[/latex]

This means that, when there is the same base and exponent that is multiplied by the same base with a different exponent, the total exponent value can be found by adding up the exponents.

[latex]\text{Product Rule of Exponents: }x^m \times x^n = x^{m+n}[/latex]

Example 6.1.2

What is the value of [latex](a^2)^3[/latex]?

[latex](a^2)^3[/latex] means that you have [latex](a^2) \times (a^2) \times (a^2)[/latex],

which is the same as [latex](a \times a) (a \times a) (a \times a)[/latex]

or [latex](a \times a \times a \times a \times a \times a)[/latex],

which equals [latex]a^6[/latex]

When you have some base and exponent where both are multiplied by another exponent, the total exponent value can be found by multiplying the two different exponents together.

[latex]\text{Power of a Power Rule of Exponents: }(x^m)^n = x^{mn}[/latex]

Example 6.1.3

What is the value of [latex](ab)^2[/latex]?

[latex](ab)^2[/latex] means that you have [latex](ab) \times (ab)[/latex],

which is the same as [latex](a \times b) \times (a \times b)[/latex]

or [latex](a \times a \times b \times b)[/latex],

which equals [latex]a^2b^2[/latex]

[latex]\text{Power of a Product Rule of Exponents: }(xy)^n = x^ny^n[/latex]

Example 6.1.4

What is the value of [latex]\dfrac{a^5}{a^3}[/latex]?

[latex]\dfrac{a^5}{a^3}[/latex] means that you have [latex]\dfrac{a \times a \times a \times a \times a}{a \times a \times a}[/latex], or that you are multiplying [latex]a[/latex] by itself five times and dividing it by itself three times.

Multiplying and dividing by the exact same number is a redundant exercise; multiples can be cancelled out prior to doing any multiplying and/or dividing. The easiest way to do this type of a problem is to subtract the exponents, where the exponents in the denominator are being subtracted from the exponents in the numerator. This has the same effect as cancelling any excess or redundant exponents.

For this example, the subtraction looks like [latex]a^{5-3},[/latex] leaving [latex]a^2.[/latex]

[latex]\text{Quotient Rule of Exponents: }\dfrac{x^m}{x^n}=x^{m-n}\hspace{0.25in} (x \ne 0)[/latex]

Example 6.1.5

What is the value of [latex]\left(\dfrac{a}{b}\right)^3[/latex]?

Expanded, this exponent is the same as:

[latex]\dfrac{a}{b}\times \dfrac{a}{b}\times \dfrac{a}{b}[/latex]

Which is the same as:

[latex]\dfrac{a \times a \times a}{b \times b \times b} \text{ or } \dfrac{a^3}{b^3}[/latex]

One can see that this result is very similar to the power of a product rule of exponents.

[latex]\text{Power of a Quotient Rule of Exponents: }\left(\dfrac{x}{y}\right)^n = \dfrac{x^n}{y^n}\hspace{0.25in} (y \ne 0)[/latex]

Simplify the following.

• [latex]4\cdot 4^4\cdot 4^4[/latex]
• [latex]4\cdot 4^4\cdot 4^2[/latex]
• [latex]2m^4n^2\cdot 4nm^2[/latex]
• [latex]x^2y^4\cdot xy^2[/latex]
• [latex](3^3)^4[/latex]
• [latex](4^3)^4[/latex]
• [latex](2u^3v^2)^2[/latex]
• [latex](xy)^3[/latex]
• [latex]4^5 \div 4^3[/latex]
• [latex]3^7 \div 3^3[/latex]
• [latex]3nm^2 \div 3n[/latex]
• [latex]x^2y^4 \div 4xy[/latex]
• [latex](x^3y^4\cdot 2x^2y^3)^2[/latex]
• [latex][(u^2v^2)(2u^4)]^3[/latex]
• [latex][(2x)^3 \div x^3]^2[/latex]
• [latex](2a^2b^2a^7) \div (ba^4)^2[/latex]
• [latex][(2y^{17}) \div (2x^2y^4)^4]^3[/latex]
• [latex][(xy^2)(y^4)^2] \div 2y^4[/latex]
• [latex](2xy^5\cdot 2x^2y^3) \div (2xy^4\cdot y^3)[/latex]
• [latex](2y^3x^2) \div [(x^2y^4)(x^2)][/latex]
• [latex][(q^3r^2)(2p^2q^2r^3)^2] \div 2p^3[/latex]
• [latex](2x^4y^5)(2z^{10}x^2y^7) \div (xy^2z^2)^4[/latex]

Answer Key 6.1

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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8th grade (Illustrative Mathematics)

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

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6.1: 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 \(10^{−n}\)

• Unit 1 - Exponents and Scientific Notation
• Unit 2 - Equations
• Unit 3 - Equations in Two Variables
• Unit 4 - Systems of Equations
• Unit 5 - Functions
• Unit 6 - Transformations
• Unit 7 - Angles
• Unit 8 - Scatter Plots
• Unit 9 - Roots and Rational Numbers
• Unit 10 - Distance Formula and Pythagorean Theorem
• Unit 11 - 3D Geometry

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