exponents problem solving worksheets

Create an unlimited supply of worksheets for practicing exponents and powers. Students can solve simple expressions involving exponents, such as 3 , (1/2) , (-5) , or 8 , or write multiplication expressions using an exponent. The worksheets can be made in html or PDF format (both are easy to print).

Each worksheet is randomly generated and thus unique. The and is placed on the second page of the file.

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This page contains links to free math worksheets for Exponents problems. Click one of the buttons below to see all of the worksheets in each set. You can also use the 'Worksheets' menu on the side of this page to find worksheets on other math topics.

Simple Exponents and Powers of Ten

20 exponents worksheets.

Practice exponents worksheets introducing exponent syntax, calculation of simple exponents, powers of ten and scientific notation. The first of these exponents worksheets include hints and will build familiarity with many common exponential terms.

Addition with Exponents

12 exponents worksheets.

Adding exponents worksheets, including simple problems where exponents are combined and order of operations rules (PEMDAS) must be observed.

Mixed Addition And Subtraction with Exponents

Worksheet practice mixing exponents with addition and subtraction operations. This is great practice for helping learn order of operations with exponents.

Multiplication with Exponents

Practice exponents worksheets mixing exponents with simple multiplication. Again, these are fantastic exercises for enhancing student comprehension of basic order of operations concepts where exponents are involved.

Powers of Ten and Scientific Notation

40 exponents worksheets.

Exponents worksheets for computing powers of ten and scientific notation, including positive exponents and negative exponents.

After learning multiplication, exponents are an important part of understanding fundamental numeric nomenclature and order of operations. The exponents worksheets in this section provide practice that reinforces the properties of exponents, including the basic procedures for adding exponents, subtracting exponents, dividing exponents and multiplying exponents. Exponents are also a critical part of understanding scientific notation, and one of the sets of exponents worksheets in this section focuses exclusively on powers of ten and exponents with base 10 to reinforce these concepts.

There are a number of simple exponent rules that make solving the problems on these exponents worksheets easier.

The value of any term with an exponent of zero is always equal to one.

The value of any term with an exponent of one is always equal to the base.

The value of any term with a negative exponent is the reciprocal of the same term with a positive exponent instead.

Multiplying Exponents

When multiplying two exponents with the same base, the result is the same as a term with base and an new exponent created by adding the two exponents in the terms of the problem.

Dividing Exponents

Similarly, for dividing exponents with the same base, the result is the same as a term with base and an new exponent created by subtracting the two numerator's exponent from the denominator's exponent.

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Use these worksheets and task cards to help students learn about exponents. Most worksheets on this page align with Common Core Standard 6.EE.1.

Exponents w/ Whole Numbers as a Base

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Exponents w/ Decimal Bases

Practice writing numbers in scientific notation.

Review squared numbers and square roots with these printables.

These worksheets can be used for teaching comparing numbers, ordering, and expanded form.

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Free Exponent Worksheets (pdfs) with answer keys

Algebra 2 exponent worksheets.

• Simplify Rational Expressions
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Algebra 1 Exponent Worksheets

Ultimate math solver (free) free algebra solver ... type anything in there, popular pages @ mathwarehouse.com.

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Exponents Worksheets

Reading and writing exponents worksheets.

Our grade 5 exponents worksheets introduce the use of exponents , including the special case of powers of ten .

Sample Grade 5 Exponents Worksheet

More exponents worksheets

Explore all of our exponents worksheets , from reading and writing simple exponents to negative exponents and equations with exponents.

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Exponents Worksheets

Exponent  is the way of defining large numbers in terms of powers. An  exponent  refers to the number of times a number is multiplied by itself.

Exponentiation  is a mathematical operation, written as  bⁿ , involving two numbers, the  base b,  and the exponent or  power n , and defined as “ b  to the power of  n “.

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Base and Exponent

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• Laws of Exponents

Laws of Exponents Worksheets

Utilize our printable laws of exponents worksheets as an essential guide for operating on problems with exponents. By steadily practicing these worksheets, students of grade 7, grade 8, and high school will be able to ace their tests in problems using the laws of exponents. Included within this set are worksheets catering to a wide range of topics such as laws of exponents, product rule, quotient rule, power of a power rule, power of a product rule, power of a quotient rule, and a few more. Now, you can access some of our worksheets for free!

Law of Exponents Chart

Explore this chart that works as a handy reference for 7th grade students to brush up their knowledge of the various and important Law of exponents. Explaining Law of exponents with crystal-clear examples, this chart helps them drive home the concept.

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Law of Exponents: Product Rule (a m *a n = a m+n )

The product rule is: when you multiply two powers with the same base, add the exponents. Train 8th grade students to rewrite each exponential expression as a single exponent with this set of pdf worksheets. Also, help them develop substantial skills in finding the value of the unknown exponent and MCQ.

• Download the set

Law of Exponents: Quotient Rule ((a m /a n ) = a m-n )

Upgrade your skills in solving problems involving quotient rule by practicing these printable worksheets. The rule states that you can divide two powers with the same base by subtracting the exponents. To facilitate easy practice with numerals and variables, the worksheets are divided into two types.

Law of Exponents: Power of a Power Rule ((a m ) n = a mn )

Look through this set of pdf worksheets to gain sufficient knowledge in rewriting an exponential expression as a single exponent form and solving an exponential equation to find the value of the unknown. Take up the mini MCQ at the end of the worksheets.

Law of Exponents: Power of a Product Rule ((a*b) m = a m *b m )

The power of a product rule states that a term raised to a power is equal to the product of its factors raised to the same power. In order that students apply the rule with confidence and their learning becomes super easy, a variety of problems involving numerals and variables are provided.

Law of Exponents: Power of a Quotient Rule ((a/b) m = (a m /b m ))

The quotient rule states that two powers with the same base can be divided by subtracting the exponents. Follow this simple rule to adeptly and quickly solve exponent problems using the power of a quotient rule. Simplify the questions by performing arithmetic operations and applying the rule.

Evaluate Using Law of Exponents: Mixed review - Type 1

Master evaluating expressions involving numerals with this set of printable worksheets that offers two levels of practice. With eight problems in every page, high school students become well versed in the concept.

Evaluate Using Law of Exponents: Mixed review - Type 2

Employ this stock of pdf worksheets to boost your practice of evaluating expressions involving numerals and variables. The key is for children to use prime factorization and apply the relevant rule of exponents. To arrive at the full and final answer, let them now perform arithmetic operations.

Finding the Missing Variable

Reiterate the concept of finding the value of the missing variable using exponent rules with these printable worksheets. Apply the law of exponents and solve for the variable x in section A; apply the law and solve for the variables x and y in section B.

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About These 15 Worksheets

These worksheets will help students practice, reinforce, and master various algebraic concepts and skills. These worksheets typically cover a wide range of topics, from basic equation solving to more complex applications involving multiple steps and different mathematical principles. They are organized to challenge students at different levels of proficiency, ensuring a gradual and thorough understanding of algebraic operations and their applications.

One common type of algebra worksheet focuses on solving one-step equations. These exercises are fundamental in algebra, as they help students understand the basic operations of addition, subtraction, multiplication, and division in the context of algebra. For example, students might be asked to solve simple equations like x + 5 = 12 or 3x = 15. The goal is to isolate the variable on one side of the equation, which reinforces the concept of balancing equations-a core idea in algebra. These worksheets typically provide practice in manipulating equations involving integers, fractions, and decimals, helping students gain confidence in their ability to solve equations quickly and accurately.

Another important category includes two-step equations and word problems. Two-step equations require students to perform two operations to solve for the variable, such as solving equations like 2x + 3 = 11. These worksheets help students understand the sequence of operations and the logic behind solving more complex equations. Word problems add an additional layer of complexity by requiring students to translate a written scenario into an algebraic equation before solving it. This not only tests their algebra skills but also their ability to interpret and model real-world situations mathematically. For instance, a problem might describe a situation where a certain number of items are purchased at different prices, and students must figure out the total cost or remaining budget.

Moving on to more advanced topics, some worksheets focus on four-step equations. These problems are designed to challenge students with equations that require multiple steps and operations, such as combining like terms, distributing, and then solving the resulting simpler equation. These types of problems help students develop critical thinking and problem-solving skills, as they must carefully consider the order of operations and the logical steps needed to isolate the variable. For example, a problem might involve an equation like 3(2x – 5) + 4 = 2x + 10, where students need to distribute, combine like terms, and then solve for x.

Worksheets on absolute value equations are another key component of algebra practice. These problems require students to understand that absolute value represents the distance from zero, regardless of direction, and therefore can result in two possible solutions for an equation. For instance, solving ∣x – 3∣ = 5 means considering both x- 3 = 5 and x – 3 = -5. These exercises help students grasp the concept of absolute value and apply it in solving equations, reinforcing their understanding of how different mathematical operations affect the solution set of an equation.

In algebra, the ability to combine like terms is crucial for simplifying expressions and solving equations. Worksheets focused on this skill typically present a variety of expressions that students must simplify by combining terms with the same variable part. For example, an expression like 2x + 3x – 5 can be simplified to 5x – 5. These exercises teach students how to identify like terms, add or subtract coefficients, and simplify expressions, which are necessary skills for more advanced algebraic manipulations.

Equations involving percentages are also common in algebra worksheets, especially as students begin to explore applications of algebra in real-world contexts. These problems often involve finding a percentage of a number, increasing or decreasing a number by a percentage, or solving for a value given its percentage relation to another value. For instance, a problem might ask students to find the percentage of students who passed an exam out of the total number of students, or to calculate the original price of an item after a discount has been applied. These exercises are particularly useful for teaching students how to apply algebraic thinking to everyday financial situations, such as shopping, budgeting, and understanding interest rates.

Logarithmic equations represent a more advanced topic in algebra, where students must apply the rules of logarithms to solve equations. These worksheets typically require students to use properties such as the product, quotient, and power rules to simplify and solve logarithmic expressions. For example, solving an equation like log(x) + log(2) = log(6) might involve applying the product rule to combine logarithms before solving for x. These exercises help students develop a deeper understanding of logarithms, which are essential for more advanced studies in mathematics and science.

Worksheets on operations with exponents and zero exponents are crucial for building a solid foundation in algebra. These exercises teach students the rules for multiplying, dividing, and raising powers to powers, as well as understanding the significance of a zero exponent. For instance, students might solve problems like (2x 3 ) (3x 2 ) or simplify expressions like x 0 . These skills are fundamental not only in algebra but also in higher-level math courses, such as calculus and physics, where exponential functions and operations frequently appear.

What Is Algebra?

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations and understand relationships between quantities. These symbols, often represented by letters such as x, y, and z, stand for numbers, and the process of algebra involves finding the values of these variables that make certain equations true. Unlike arithmetic, which deals with specific numbers and straightforward calculations, algebra introduces the concept of generalization, allowing for the expression of complex relationships in a more abstract form. This abstraction enables algebra to solve problems that are not confined to specific numerical cases, making it a powerful tool in mathematics.

Algebra is used in a wide array of fields and everyday situations. In science and engineering, algebra is essential for modeling and solving problems that involve relationships between varying quantities, such as the motion of objects, electrical circuits, or chemical reactions. In economics, algebra helps in understanding and predicting changes in markets, calculating interest rates, and optimizing business operations. In computer science, algebraic principles underpin algorithms, data structures, and cryptography. Even in daily life, algebra is often used, whether in calculating a budget, adjusting a recipe, or determining the amount of paint needed for a room. Algebra’s ability to generalize and simplify problems makes it an indispensable tool for problem-solving in both academic and practical contexts.

Powers and Exponents Worksheets

Powers and exponents worksheets will produce power and exponents problems for practicing power and exponents problems calculations.These worksheets are very helpful for Kids for basic knowledge. Worksheets give students the opportunity to solve a wide variety of problems helping them to build a robust mathematical foundation.

Benefits of Powers and Exponents Worksheets

Powers and exponents worksheets can help students to understand the concept of power and how to apply it to solve more complex math problems that have multiple terms and multiple operations.

Powers and exponents worksheets are interactive and provide several visual simulations. Hence, they enable students to combine fun with learning, creating a positive learning environment. They are also easy to use and free to download.

Powers and exponents worksheets help students off with familiar concepts of division, multiplication, addition and subtraction before adding parentheses and exponents.

Download Powers and Exponents Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

☛ Check Grade wise Exponents Worksheets

• 5th Grade Exponents Worksheets
• Exponents Worksheets 6th Grade
• 7th Grade Exponents Worksheets
• Exponents Worksheets for Grade 8

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Solving a System of Equations Word Problems Worksheet

System of Equations is the combination of equations containing variables. For example, 2x+3y= 5 and 3x+2y= 10 are systems of equations.

In this article, we will learn how to solve a System of Equations.

Table of Content

What is a System of Equations?

Important concepts, solved problems, worksheet: system of equations.

A system of equations is a collection of two or more equations involving the same set of variables. The objective is to find the values of these variables that satisfy all the equations simultaneously. Systems of equations are used in various fields, including mathematics, physics, engineering, economics, and more.

Example: A movie theater sells two types of tickets: adult tickets for $12 each and child tickets for $8 each. On a particular day, the theater sells a total of 150 tickets and earns $1,500 in revenue. How many adult tickets and child tickets were sold?

Let x be the number of adult tickets sold. Let y be the number of child tickets sold. Define the variables: Set up the system of equations: The total number of tickets sold is 150: x+y=150(Equation 1) The total revenue is $1,500: 12x+8y=1500(Equation 2) Solve the system using the substitution method: Solve Equation 1 for y:y=150−x Substitute this expression for y in Equation 2:12x+8(150−x)=1500 Distribute and combine like terms:12x+1200−8x=1500 4x + 1200 = 1500 Solve for x:4x=300⇒x=75 Now, substitute x=75 back into Equation 1 to find y:y=150−75=75 Answer: The theater sold 75 adult tickets and 75 child tickets. Verification: Total tickets: 75+75=150 (Matches Equation 1) Total revenue: 12(75)+8(75)=900+600=1500(Matches Equation 2)

Some important concepts related to solving a system of equations are listed below:

Types of Systems

• Linear System: All equations in the system are linear, i.e., each equation can be written in the form ax+by+cz+⋯=d.
• Non-Linear System: At least one equation is non-linear, such as quadratic, exponential, or trigonometric equations.

Number of Solutions

• Consistent System: The system has at least one solution.
• Inconsistent System: The system has no solutions.

Problem 1: A fruit seller sells apples and oranges. The price of 3 apples and 4 oranges is $22. The price of 2 apples and 3 oranges is $16.

Let the price of one apple be x and the price of one orange be y. We have the following system of equations: 3x+4y=22(Equation 1) 2x+3y=16(Equation 2) Step 1: Multiply Equation 1 by 2 and Equation 2 by 3 to eliminate x:6x+8y=44(Equation 3) 6x+9y=48(Equation 4) Step 2: Subtract Equation 3 from Equation 4:(6x+9y)−(6x+8y)=48−44 y=4 Step 3: Substitute y=4 into Equation 1:3x+4(4)=22 3x+16= 22 x=6⇒x=2 Answer: The price of one apple is $2, and the price of one orange is $4.

Problem2 : A father is 3 times as old as his son. In 10 years, he will be twice as old as his son.

Let the father’s current age be x and the son’s current age be y. We have the following system of equations: x=3y(Equation 1) x+10=2(y+10)(Equation 2) Step 1: Substitute Equation 1 into Equation 2:3y+10=2(y+10)3 3y + 10 = 2y + 20 Step 2: Solve for y:y=10 Step 3: Substitute y=10 into Equation 1:x=3(10)=30 Answer: The father is 30 years old, and the son is 10 years old.

Problem 3: Two trains are moving towards each other from two stations 300 miles apart. Train A moves at 60 mph and Train B at 40 mph.

Let t be the time in hours when they meet. We have the following equation: 60t+40t=300 Step 1: Combine like terms:100t=300 Step 2: Solve for t:t=3 hours Answer: The trains will meet in 3 hours.

Problem 4: A store sells two types of pens. One costs $2 each and the other $3 each. A customer buys a total of 15 pens for $37.

Let the number of $2 pens be x and the number of $3 pens be y. We have the following system of equations: x+y=15(Equation 1) 2x+3y=37(Equation 2) Step 1: Solve Equation 1 for y:y=15−x Step 2: Substitute into Equation 2:2x+3(15−x)=37 2x + 45 – 3x = 37 Step 3: Solve for x:−x+45=37 ⇒x=8 Step 4: Find y:y=15−8=7 Answer: The customer bought 8 $2 pens and 7 $3 pens.

Problem 5: A farmer has 100 feet of fencing and wants to enclose a rectangular area along the river, using the river as one side of the rectangle.

Let the length of the rectangle be x and the width be y. We have the following system of equations:2x+y=100(Equation 1) Step 1: Solve for y:y=100−2xy = 100 – 2xy=100−2x Step 2: Express the area A as a function of x:A=x⋅y=x(100−2x)=100x−2x 2 Step 3: To maximize A, find the vertex of the parabola:x=−b 2 /a=−100/−4=25 Step 4: Find y:y=100−2(25)=50 Answer: The dimensions that maximize the area are 25 feet by 50 feet.

Problem 6: A coffee shop sells a blend of coffee made from two types of beans. Type A costs $5 per pound, and Type B costs $8 per pound. The shop sells a 10-pound mixture for $65.

Let the number of pounds of Type A be x and Type B be y. We have the following system of equations: x+y=10(Equation 1) 5x+8y=65(Equation 2) Step 1: Solve Equation 1 for y:y=10−x Step 2: Substitute into Equation 2:5x+8(10−x)=65 5x + 80 – 8x = 65 Step 3: Solve for x:−3x+80=65⇒x=5 Step 4: Find y:y=10−5=5 Answer: The mixture contains 5 pounds of Type A and 5 pounds of Type B.

Problem 7: A person invests $10,000 in two accounts, one with a 5% interest rate and the other with a 7% interest rate. The total interest earned in a year is $640.

Let the amount invested at 5% be x and the amount at 7% be y. We have the following system of equations: x+y=10000(Equation 1) 0.05x+0.07y=640(Equation 2) Step 1: Solve Equation 1 for y:y=10000−xy = 10000 – xy=10000−x Step 2: Substitute into Equation 2:0.05x+0.07(10000−x)=640 .05x+700−0.07x=640 Step 3: Solve for x:−0.02x+700=640⇒x=300 Step 4: Find y:y=10000−3000=7000 Answer: $3000 was invested at 5%

Problem 1: A boat travels 30 miles upstream in 4 hours and returns downstream in 3 hours. What are the speed of the boat in still water and the speed of the current?

Problem 2: Two planes leave the same airport, one heading north at 450 mph and the other heading west at 600 mph. After 2 hours, how far apart are they?

Problem 3:A man buys 5 pencils and 3 pens for $2.50. The next day, he buys 2 pencils and 7 pens for $3.50. Find the cost of each pencil and pen.

Problem 4: Two cyclists start from the same point and travel in the same direction. One travels at 15 mph and the other at 12 mph. After how many hours will they be 9 miles apart?

Problem 5: A farmer has 50 chickens and cows. The total number of legs is 140. How many chickens and cows does he have?

Problem 6: A theater sells tickets for $10 and $15. If 300 tickets were sold and the total revenue was $3500, how many tickets of each type were sold?

Problem 7: A student has a collection of 100 coins consisting of nickels and dimes. The total value of the coins is $7.50. How many nickels and dimes does the student have?

Problem 8: A car rental company charges a flat fee of $50 and $0.20 per mile driven. If a customer pays $70, how many miles did they drive?

Problem 9: Two cars leave the same point, one traveling north at 60 mph and the other east at 80 mph. After 3 hours, how far apart are the cars?

Problem 10: A chemist needs to mix two solutions, one containing 10% acid and the other containing 20% acid, to make 100 liters of a solution containing 15% acid. How many liters of each solution should be used?

What is a system of equations?

A system of equations is a set of two or more equations with the same variables. The goal is to find values for the variables that satisfy all the equations in the system simultaneously.

When is the graphing method useful?

The graphing method is useful when you want a visual representation of the solutions. It’s particularly helpful when the system involves simple linear equations or when approximate solutions are acceptable.

What is a system of nonlinear equations?

A system of nonlinear equations involves at least one equation that is not linear (e.g., quadratic, exponential). Solving these systems may require more advanced methods such as substitution, graphing, or numerical methods.

Can a system of equations have more than two variables?

Yes, a system of equations can have three or more variables. These systems can be solved using similar methods, but they may involve more complex algebra or the use of matrices and determinants.

What is a homogeneous system of equations?

A homogeneous system of equations is one where all the constant terms are zero (e.g., ax+by=0ax + by = 0). Such systems always have at least one solution (the trivial solution where all variables are zero).

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