Math Core

Writing $2 \times 2 \times 2 \times 2 \times 2 \times 2$ takes a while, and it's easy to lose count of the twos. Exponents give you a short way to write repeated multiplication, and they show up everywhere from square floor tiles to computer memory.

Repeated multiplication

You already know that multiplication is a shortcut for repeated addition: $4 + 4 + 4 = 3 \times 4$ . An exponent does the same job for repeated multiplication.

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

Definition

Exponent

In $b^n$ , the number $b$ is the base and the small raised number $n$ is the exponent. The exponent tells you how many times the base is used as a factor:

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

The whole expression $b^n$ is called a power. You read $2^6$ as "2 to the sixth power."

So $2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$ .

Squares and cubes

Two powers have special names because of their connection to shapes.

$5^2$ is read "5 squared." A square with sides of 5 units has an area of $5 \times 5 = 25$ square units.
$5^3$ is read "5 cubed." A cube with edges of 5 units has a volume of $5 \times 5 \times 5 = 125$ cubic units.

A square with side 3 is made of 3 × 3 = 9 unit squares, so its area is 3² = 9 square units.

Evaluating powers

To find the value of a power, write out the factors and multiply from left to right.

Worked example: Whole-number bases

Evaluate $3^4$ and $10^3$ .

3^4 = 3 \times 3 \times 3 \times 3 = 9 \times 3 \times 3 = 27 \times 3 = 81

10^3 = 10 \times 10 \times 10 = 1{,}000

Tip

Powers of 10 are easy: the exponent tells you how many zeros follow the 1. So $10^5 = 100{,}000$ .

Common mistake

The exponent is not a number you multiply by. $4^3$ means $4 \times 4 \times 4 = 64$ , not $4 \times 3 = 12$ . Always write out the factors if you are unsure.

Fractions and decimals as bases

The base can be a fraction or a decimal. The rule is the same: multiply the base by itself.

Worked example: A fraction base

Evaluate $\left(\dfrac{2}{3}\right)^2$ .

\left(\dfrac{2}{3}\right)^2 = \dfrac{2}{3} \times \dfrac{2}{3} = \dfrac{2 \times 2}{3 \times 3} = \dfrac{4}{9}

The parentheses show that the whole fraction is the base, so both the top and the bottom get squared.

Worked example: A decimal base

Evaluate $(0.3)^3$ .

0.3 \times 0.3 = 0.09, \qquad 0.09 \times 0.3 = 0.027

So $(0.3)^3 = 0.027$ . Notice that the answer is smaller than $0.3$ . Multiplying by a number less than 1 makes things smaller.

Writing with exponents

You can also go the other way and rewrite a product using an exponent. Count how many times the base appears.

Worked example: From product to power

Write $7 \times 7 \times 7 \times 7 \times 7$ using an exponent. Then write $81$ as a power of $3$ .

The factor $7$ appears 5 times, so the product is $7^5$ .
Keep multiplying by 3 until you reach 81: $3 \times 3 = 9$ , $9 \times 3 = 27$ , $27 \times 3 = 81$ . That used four 3s, so $81 = 3^4$ .

One more fact: any number to the first power is just that number, because it is used as a factor only once. So $9^1 = 9$ .