Evaluating and Understanding Exponents

We start with the basic nature of exponents. How to convert between standard form and other forms of notation.

1. Compare Exponents | Answer Key - We cover a wide range of skills in this printable. It starts very basic and gets more comprehensive.
2. Complete The Equations | Answer Key - This is a really good introduction to this skill. We walk students towards the visual understanding of exponents.
3. Evaluate Exponents | Answer Key - This entire series gets progressively more difficult with each problem. If students can breeze through this one, they have a good handle on exponents.
4. Evaluate Exponents to a Single Number | Answer Key - This worksheet asks students to simplify the expression down to one number. As usually it gets progressively harder.
5. Simplify Exponents | Answer Key - This covers a wide range of like term procedures. It also covers exponential operations very well.
6. Version 2 | Answer Key
7. Version 3 | Answer Key
8. Version 4 | Answer Key
9. Using Scientific Notation | Answer Key - Convert numbers in standard form to scientific notation. Note the use of decimals and the negative sign that needs to be used.
10. Version 2 | Answer Key
11. Standard Notation | Answer Key - You are provided with numbers in scientific notation. Convert them to standard form. The negative symbols can throw students off.

Working With Exponents

Students work with exponents in different situations. It is great for intermediate students. We work on the many different operations that exponents can be used in.

1. Division of Monomials | Answer Key - This one really turns up the heat on the use of like variables. This usually intimidates students. We suggest doing the first problem with them.
2. Multiply Monomials Worksheet | Answer Key - A very challenging set of problems for students. Remind them to check their work every step of the way.
3. Multiply Monomials with Negatives | Answer Key - Make sure to pay attention to the sign of each integer as a whole number and an exponent. Many times students will overlook this.
4. Multiply Monomials with Negatives and Powers Worksheet | Answer Key
5. Simplify Complex Expressions | Answer Key - Perform a wide variety of operations on a series of monomials. This should be one of the first operation type activities they perform.
6. Operations With Scientific Notation | Answer Key - This is an interesting one. Scientists are constantly performing these types of operations.
7. Version 2 | Answer Key
8. Version 3 | Answer Key
9. Version 4 | Answer Key

Converting Exponent Worksheets

These sheets work on standard conversion between exponents and whole numbers.

1. Convert Exponents Version 1 - A simple conversion activity of exponents. We do stress the use of ones and zeros in exponents.
2. Convert Version 1 Answers


3. Convert Version 2 - We use negative exponents as well.
4. Convert Version 2 Answers


5. Convert Version 3
6. Convert Version 3 Answers

Order of Operations with Exponents Worksheets

1. Comparing Exponents - Have students compare the end value of these exponents.
2. Operations with Exponents - We take exponents out for a walk and throw them into 2 and 3 step problems.
3. Order Of Operations With Exponents 1 - This is a really big one. 10 printable pages in all. 3 versions that are well spaced out for you.
4. Order of Operations with Parenthesis and Exponents - The go deep on this one. Just about every level skill is covered here.

Exponents - Definition, Types, and Rules

A number's exponent shows the number of times we multiply a digit by itself. For instance, 56 implies that we are multiplying the number 5, six times. If we expand 56 it would be 5 x 5 x 5 x 5 x 5 x 5.

Exponents are often referred to as the number's power. It could be an integer, a fraction, decimals, or a negative number.

What are They?

An exponent is an equation that comprises a repeating multiplication power with the same variable. It is also known as indices or raised to the power of a value

Let's look at the following example:

3⁴

Here, the number 3 is known as the base of the exponent, while 4 is the exponent or the power of the equation.

The result of the equation is obtained by multiplying its base the same number of times as the number in the exponent's place. For the above-mentioned example, the exponent is 4, therefore the value will become 3 x 3 x3 x 3 = 81

Types of Exponents

Depending on the number inside the power, exponents are classified into four categories. These categories are as follows:

1. Positive
2. Zero
3. Negative
4. Rational

Positive

Positive exponents can always be shortened easily by multiplying the base with itself the same number of times as the exponent.

For instance:

2⁶ = 2 x 2 x 2 x 2 x 2 x 2 = 64
5⁴ = 5 x 5 x 5 x 5 = 625

Zero

Any equation with the exponent or power of 0 will become 1, and there is no need to take into account the base value for simplification.

5⁰ = 1
500⁰ = 1

Negative

A negative power can be shortened by putting 1 in the numerator of the base and moving the expression to the denominator. Doing so will flip the sign of the power in the denominator.

Let's look at a few examples:

3^-4 = 1 / 3⁴ = 1/81 = 0.0123

2^-8 = 1/2⁸ = 1/256 = 0.0039

Rational Forms

Exponents that are fractional or rational will turn into roots or radicals. For instance, 8^1/3 can be will be of cube root 8, 9^3/2 will be written as the square root of 9 raised to the power 3.

Let's solve these:

8^1/3 = 3√ 8 = 2

The answer to the expression 8^1/3 is 2.

9^3/2 = 2√9³

2√ 9 = 3

3³ = 3 x 3 x 3 = 27

The answer to the expression 9^3/2 is 27.

Rules of Exponents

There are some rules of exponents for solving equations with exponents. Ensure you extensively review each rule of exponent because each one is vital in answering exponent-based problems.

Product Rule (same exponent)

According to this rule, if the exponent is the same for multiple distinct bases, then;

a^c x b^c = (ab)^c

For instance: 5² x 6²

(5 x 6)² = 302

= 900

Product Rule (same base)

When multiplying 2 bases with the exact same number, leave the bases the same though add the distinct powers together and get the answer.

a^b x a^c = a^{b + c}

For instance:

3³ x 3²

3^{3 + 2} = 35

= 243

Quotient Rule (same exponent)

A quotient is just the result of dividing two values. You are raising the two values that are to be divided by power in this rule. The power in an equation like this must be spread to all numbers within the parentheses it is tied to.

(a/b)^c = a^c / b^c

For instance:

(2/5)³ = 2³ / 5³

= 2 x 2 x 2 / 5 x 5 x5

= 8 / 125

=0.064

Quotient Rule (same base)

Division and multiplication are the exact opposite of each other. Similarly, the quotient rule is the inverse of the product rule.

Keep the base constant when dividing the two numbers of the same value, and afterward subtract the power values.

a^b / a^c = a^{b -c}

For instance:

5⁵ / 5² = 5^{5 -2}

= 5³ = 5 x 5 x 5 = 125

Power Rule

This rule explains how to solve equations in which one exponent raises another. With such equations, you will multiply the powers present with each other while keeping the base constant.

(a^b)^c= a^{b x c}

For instance:

(3²)³ = 3^{2 x 3}

= 3⁶ = 3 x 3 x 3 x 3 x 3 x 3

= 729

Power and Radical Rule

If there is an expression with an exponent inside a radical, we'll bring the radical to the denominator of the power

a √ b^c = b^c/a

For instance:

5 √4¹⁰ = 4^10/5

= 10 / 5 = 2

= 4² = 4 x 4 = 16

Ones Rule (exponents)

If the exponent of a number is one the base will remain unchanged no matter how big the number within it is.

a¹ = a

For instance:

5¹ = 5

1250¹ = 1250

One Rule (base)

If the base of the number is one, no matter what the power is, the base will remain the same.

1^a = 1

For instance:

1³ = 1 x 1 x 1 = 1
1¹⁰= 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 = 1

Wrapping Up

Exponents are an integral part of mathematics. They not only help simplify mathematical equations, but they also aid in developing quick mental math skills.

All you need to do is familiarize yourself with all the rules of exponents and understand how different types of exponents work, and you'll be all set to solve any exponential expression.

Exponents Related Teacher Resources

1. Exponent Lesson Plans
2. Teacher Curriculum For Math