What Are Exponents? Powers, Bases and Repeated Multiplication

Q: What is the difference between an exponent and a power?

People use both words loosely. Strictly, in 2^5 = 32 the exponent is 5 and 32 is a power of 2. In everyday use, "to the power 5" and "exponent 5" mean the same thing.

What $b^n$ means, why squares and cubes are named after shapes, how quickly powers grow, and the bracket rule behind $(-3)^2 = 9$ but $-3^2 = -9$ .

4 min read Supports 3 lessons

Multiplication is a shortcut for repeated addition: $4 \times 3$ means $3 + 3 + 3 + 3$ . Exponents are a shortcut for repeated multiplication: $3^4$ means $3 \times 3 \times 3 \times 3$ .

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

$b$ is the base; $n$ is the exponent, also called the power or index.

We read $b^n$ as " $b$ to the power $n$ " or " $b$ to the $n$ th".

Squares and cubes

Two powers have their own names, and the names come from geometry. A square with side $3$ is made of $3^2 = 9$ unit squares. A cube with side $3$ holds $3^3 = 27$ unit cubes.

$3^{3} = 3 \times 3 \times 3 = 27$

$3^1$ is a row, $3^2$ a square, $3^3$ a cube of $27$ blocks.

How fast powers grow

Each extra power multiplies by the base again, so powers grow astonishingly fast. Doubling $1$ thirty times gives over a billion.

a_{1}

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

128

a_{7}

Powers of two: $2^1$ to $2^7$ . Each bar is double the last.

$2^{10} = 1024$
Ten doublings: about a thousand.

1/3

Powers of ten

Powers of ten are especially friendly: $10^n$ is a $1$ followed by $n$ zeros. That is why they are used to write very large and very small numbers in scientific notation.

10^1 = 10 \qquad 10^2 = 100 \qquad 10^3 = 1000 \qquad 10^6 = 1\,000\,000

Brackets and negative bases

An exponent applies only to the thing it is attached to. That makes brackets important when the base is negative.

(-3)^2 = (-3)(-3) = 9 \qquad\qquad -3^2 = -(3 \times 3) = -9

Watch out

$-3^2 = -9$ . Without brackets, the power is worked out first and the minus sign is applied afterwards, following the order of operations.

Tip

A negative number raised to an even power is positive; to an odd power it is negative. $(-2)^4 = 16$ but $(-2)^3 = -8$ .

Watch out

$2^3$ is not $2 \times 3$ . It is $2 \times 2 \times 2 = 8$ .

Exponents in algebra

The same notation works with letters: $x^2$ means $x \times x$ , and $5x^3$ means $5 \times x \times x \times x$ . Notice that in $5x^3$ only the $x$ is cubed, not the $5$ .

Worked example

Evaluate $2x^3$ when $x = -2$ .

1
Substitute, with brackets.
$2(-2)^3$

Common questions

What is the difference between an exponent and a power?: People use both words loosely. Strictly, in $2^5 = 32$ the exponent is $5$ and $32$ is a power of $2$ . In everyday use, "to the power 5" and "exponent 5" mean the same thing.
Why is a negative number squared positive?: Squaring multiplies the number by itself, and a negative times a negative is positive. So $(-4)^2 = 16$ .

Practise exponents

The Powers & Exponents lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

What is the difference between an exponent and a power?

People use both words loosely. Strictly, in $2^5 = 32$ the exponent is $5$ and $32$ is a power of $2$ . In everyday use, "to the power 5" and "exponent 5" mean the same thing.

Why is a negative number squared positive?

Squaring multiplies the number by itself, and a negative times a negative is positive. So $(-4)^2 = 16$ .