Exponents Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Exponents.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An operation representing repeated multiplication: $a^n$ means $a$ multiplied by itself $n$ times. For example, $2^3 = 2 \times 2 \times 2 = 8$ . Exponents extend to zero, negative, and fractional powers.

$2^3$ means $2 \times 2 \times 2 = 8$ . The exponent tells you how many times to multiply.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An exponent counts how many times the base is used as a factor.

Common stuck point: The procedure for exponents is the easy part; the trap is multiplying base by exponent. Asking "Is the base being used as a factor again and again?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the base being used as a factor again and again?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Exponents Mistakes

Worked Examples

Example 1

easy

Compute

5^3

Answer

125

First step

Recognize that the exponent 3 means multiply the base 5 by itself three times.

Full solution

2
Write the base 5 multiplied by itself 3 times: $5 \times 5 \times 5$ .
3
Compute step by step: $5 \times 5 = 25$ , then $25 \times 5 = 125$ .

An exponent tells you how many times to multiply the base by itself. Here

5^3

means three factors of 5.

Example 2

medium

Simplify

(-2)^4

Example 3

medium

Simplify

\frac{2^5 \times 2^3}{2^4}

Example 4

medium

Simplify

\frac{x^7}{x^3}

Example 5

medium

Simplify

(2x^2)^3

Example 6

medium

Simplify

\frac{6x^5}{2x^2}

Example 7

hard

Simplify

\frac{(2x^3)^2 \cdot x^4}{x^5}

Example 8

hard

Simplify

8^{2/3}

Example 9

hard

Solve

3^{x+1} = 81

Example 10

challenge

Simplify

\frac{(x^2 y^{-3})^{-2}}{x^{-1} y^4}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Compute

2^6

Example 2

medium

Which is larger,

3^4

4^3

? Compute both to compare.

Example 3

easy

Compute

3^4

Example 4

easy

Compute

5^0

Example 5

easy

Compute

2^5

Example 6

easy

Compute

10^3

Example 7

easy

Compute

2^{-3}

Example 8

easy

Compute

4^2 \cdot 4^3

as a power of

4

Example 9

easy

Compute

\frac{7^5}{7^3}

Example 10

easy

Compute

(3^2)^3

Example 11

medium

Compute

2^3 \cdot 2^4 \cdot 2^{-2}

Example 12

medium

Compute

\frac{6^7}{6^4}

Example 13

medium

Compute

(2 \cdot 3)^4

Example 14

medium

Compute

\left(\frac{2}{3}\right)^3

Example 15

medium

Simplify

\frac{x^5 \cdot x^3}{x^2}

Example 16

medium

Compute

9^{1/2}

Example 17

medium

Compute

8^{2/3}

Example 18

medium

Simplify

(2x^3)^4

Example 19

medium

Compute

0.1^4

Example 20

challenge

Simplify

\frac{2^{10} \cdot 3^5}{2^7 \cdot 3^3}

Example 21

challenge

What is the units digit of

7^{2024}

Example 22

challenge

2^x = 32

, find

x

Example 23

easy

Compute

6^2

Example 24

easy

Compute

2^7

Example 25

easy

Compute

(-3)^2

Example 26

easy

Compute

(-2)^3

Example 27

easy

Compute

10^4

Example 28

easy

Compute

3^0

Example 29

medium

Simplify

(x^3)^4

Example 30

medium

Compute

9^{1/2}

Example 31

medium

Simplify

\left(\frac{x^4}{x}\right)^2

Example 32

medium

Compute

\left(\tfrac{1}{2}\right)^4

Example 33

hard

Solve for

x

2^x = 32

Example 34

hard

Compute

25^{3/2}

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Related Concepts

Multiplication Square Roots Logarithm Exponential Function

Background Knowledge

These ideas may be useful before you work through the harder examples.

multiplication