Exponents: What They Mean and How to Evaluate

Exponents are a short way to write repeated multiplication. The exponent tells how many times to multiply the base by itself. In $2^4$ , the base is 2 and the exponent is 4, so $2^4 = 2 \times 2 \times 2 \times 2 = 16$ . You read it as “two to the fourth power.”

Before this lesson: You should know how to multiply integers and how to multiply in operations with decimals.

Extra practice: The Pre-Algebra Unit 4 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

What Exponents Mean

Multiplication is repeated adding: $3 \times 5 = 5 + 5 + 5$ . In the same way, an exponent is repeated multiplying.

The base is the number being multiplied.
The exponent is the small raised number. It counts how many times the base is used.
The whole expression, like $2^4$ , is called a power.

\underbrace{2 \times 2 \times 2 \times 2}_{4 \text{ factors}} = 2^4 = 16

A power of 2 is called squared, and a power of 3 is called cubed. These names come from shapes.

On the left, a square made of 3 rows of 3 small squares shows 3 squared = 3 × 3 = 9. On the right, a cube made of 2 layers of 2 by 2 small cubes shows 2 cubed = 2 × 2 × 2 = 8. — A power of 2 makes a square, and a power of 3 makes a cube. That is why we say squared and cubed.

The Rule for Evaluating Exponents

a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ factors}}

Here $a$ is the base and $n$ is a whole number. To evaluate a power, write out the factors and multiply. Any number to the first power is itself, so $5^1 = 5$ .

Worked Examples

Example 1: Evaluate a power

Evaluate $5^3$ .

Step 1. The base is 5 and the exponent is 3, so use 5 as a factor 3 times.

5 \times 5 = 25

Step 2. Multiply the first two factors.

25 \times 5 = 125

Step 3. Multiply by the last factor of 5.

Answer: $5^3 = 125$ .

Example 2: Write repeated multiplication as a power

Write $7 \times 7 \times 7 \times 7$ as a power. Then evaluate it.

7^4

Step 1. The factor 7 appears 4 times, so the base is 7 and the exponent is 4.

Step 2. Multiply: $7 \times 7 = 49$ , $49 \times 7 = 343$ and $343 \times 7 = 2401$ .

Answer: $7^4 = 2401$ . You read it as “seven to the fourth power.”

Example 3: A negative base

Evaluate $(-3)^4$ and $-3^4$ .

(-3)(-3)(-3)(-3) = 81

Step 1. In $(-3)^4$ , the parentheses make −3 the base. Four negatives make a positive.

Step 2. In $-3^4$ , only the 3 is the base. Find $3^4 = 81$ , then take the opposite: −81.

Answer: $(-3)^4 = 81$ and $-3^4 = -81$ .

Example 4: Fraction and decimal bases

Evaluate $\left(\frac{2}{3}\right)^2$ and $0.3^2$ .

\frac{2}{3} \times \frac{2}{3} = \frac{4}{9}

Step 1. Multiply the fraction by itself. The exponent applies to the top and the bottom.

0.3 \times 0.3 = 0.09

Step 2. Multiply the decimal by itself. Count two decimal places in the answer.

Answer: $\left(\frac{2}{3}\right)^2 = \frac{4}{9}$ and $0.3^2 = 0.09$ .

Example 5: Word problem

A single cell splits into 2 cells every hour. Each new cell splits the same way. How many cells are there after 6 hours?

Step 1. The number of cells doubles each hour, so after 6 hours it has been multiplied by 2 six times.

2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64

Step 2. Write it as a power.

Answer: There are 64 cells after 6 hours.

Common Mistakes

Mistake	What it looks like	How to fix it
Multiplying the base by the exponent	$3^4 = 12$	Use the base as a factor 4 times: $3 \times 3 \times 3 \times 3 = 81$ .
Switching the base and exponent	$2^3 = 9$	$2^3 = 8$ , but $3^2 = 9$ . The base is the big number.
Ignoring the parentheses	$-3^4 = 81$	Without parentheses, only the 3 is raised: $-3^4 = -81$ .
Raising only the top of a fraction	$\left(\frac{2}{3}\right)^2 = \frac{4}{3}$	Square the top and the bottom: $\frac{4}{9}$ .
Miscounting zeros	$10^4 = 1000$	$10^4$ has 4 zeros: 10,000.

Powers of 10 and Order of Operations

Powers of 10 are easy: the exponent tells you how many zeros to write after the 1.

Power	$10^1$	$10^2$	$10^3$	$10^6$
Value	10	100	1,000	1,000,000

Powers of 10 are the key to scientific notation, which comes later in this unit.

In the order of operations, exponents come right after parentheses. So $2 + 3^2 = 2 + 9 = 11$ , but $(2 + 3)^2 = 5^2 = 25$ . Squares like 9, 16 and 25 are also called perfect squares, which you’ll use in square roots.

Rules for multiplying and dividing powers, and for zero and negative exponents, are taught in Algebra 1 (coming soon).

Practice Problems

Evaluate $4^3$ .
Evaluate $2^5$ .
Evaluate $9^2$ .
Write $10^5$ as a whole number.
Write $3 \times 3 \times 3 \times 3 \times 3$ as a power. Then evaluate it.
Evaluate $(-2)^5$ .
Evaluate $-6^2$ .
Evaluate $\left(\frac{1}{2}\right)^3$ .
Evaluate $0.2^3$ .
In a phone tree, each person calls 3 new people. In round 1, one person makes 3 calls. In each new round, everyone who was just called makes 3 calls. How many calls are made in round 4?

Show answers

4 \times 4 \times 4 = 64

Use 4 as a factor 3 times.

2 \times 2 \times 2 \times 2 \times 2 = 32

Use 2 as a factor 5 times.

9 \times 9 = 81

Use 9 as a factor 2 times.

100,000. There are 5 zeros.

3^5 = 243

There are five 3s, so write it with exponent 5 and evaluate.

(-2)^5 = -32

Five negatives make a negative.

-6^2 = -(6 \times 6)

Only the 6 is squared.

-(6 \times 6) = -36

Multiply.

\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}

Use one-half as a factor 3 times.

0.2 \times 0.2 \times 0.2 = 0.008

Use 0.2 as a factor 3 times.

10.

3^4 = 81

The calls go 3, 9, 27, 81. Round 4 is the 4th power of 3.

Answer: 81 calls.

Want more practice? The Pre-Algebra Unit 4 Practice Pack has 35 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Exponents FAQ

What is a number to the first power?

Any number to the first power is just that number, because it is used as a factor only once. For example, $8^1 = 8$ .

What is a number to the zero power?

Any nonzero number to the zero power equals 1, so $5^0 = 1$ . The reason is explained with the exponent rules in Algebra 1.

Why is a power of 2 called squared?

A square with sides of 3 units is made of $3^2 = 9$ small squares. In the same way, a cube with sides of 2 units holds $2^3 = 8$ small cubes.

Is (−2)² the same as −2²?

No. $(-2)^2 = 4$ because the whole −2 is squared. But $-2^2 = -4$ because only the 2 is squared.

Summary

An exponent tells how many times to use the base as a factor.
$a^n$ means multiply $a$ by itself $n$ times.
A power of 2 is “squared” and a power of 3 is “cubed.”
Use parentheses for a negative base: $(-3)^4 = 81$ , but $-3^4 = -81$ .
$10^n$ is a 1 followed by $n$ zeros.

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Next lesson: Square Root

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