What Are Exponents? Powers and Bases Explained

Shorthand for repeated multiplication

$$2^5 = \underbrace{2 \times 2 \times 2 \times 2 \times 2}_{\text{five 2s}} = 32$$

The bottom number is the base — the thing being multiplied. The small raised number is the exponent — how many of them there are.

Notation	Name	Means
$2$	base	what gets multiplied
$5$	exponent, index or power	how many times

Writing $2^5$ rather than $2 \times 2 \times 2 \times 2 \times 2$ is the entire point. It gets shorter as the number gets bigger, and $2^{100}$ would otherwise be unwritable.

Why exponents are worth having

Fold a sheet of paper in half. Two layers. Fold it again — four. Each fold doubles what came before, so after $n$ folds there are $2^n$ layers.

Folds	Layers
$5$	$32$
$10$	$1024$
$20$	$1{,}048{,}576$
$30$	over a billion

Thirty folds is a stack of paper that would reach past the edge of space, and nothing in that sum was ever bigger than a doubling. That is what exponents describe: quantities that grow by repeatedly multiplying, which arrive at enormous numbers far faster than repeated adding ever does.

The notation earns its keep twice over. It keeps $2^{30}$ down to three characters, and it makes the pattern readable — you can see at a glance that $2^{30}$ is a thousand times $2^{20}$, which is invisible in $1{,}073{,}741{,}824$.

The error that matters most

$$2^5 = 32 \qquad\text{but}\qquad 2 \times 5 = 10$$

An exponent is repeated multiplication, not repeated addition. Reading $2^5$ as “two fives” gives $10$, which is not close.

The gap widens fast, which is why the mistake never stays small:

$n$	$2 \times n$	$2^n$
$3$	$6$	$8$
$5$	$10$	$32$
$10$	$20$	$1024$

Saying them aloud

Power	Said
$5^2$	five squared
$5^3$	five cubed
$5^4$	five to the fourth power
$5^1$	five

“Squared” and “cubed” are not arbitrary. $s^2$ is the area of a square with side $s$, and $s^3$ is the volume of a cube with edge $s$. The names come from the shapes.

Powers of ten

These come up constantly, and the pattern is worth seeing rather than memorising.

$$10^1 = 10 \qquad 10^2 = 100 \qquad 10^3 = 1000$$

The exponent counts the zeros. That is not a coincidence: each extra factor of ten shifts every digit one place left, which is exactly what place value does.

Where they sit in order of operations

Exponents come straight after brackets, before multiplication and division.

$$3 \times 2^4 = 3 \times 16 = 48$$

Multiplying first would give $6^4 = 1296$, which is a different question.

Brackets and negatives

This distinction is small on the page and large in the answer.

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

In the first, the brackets put the minus inside, so it gets squared too and two negatives make a positive. In the second there are no brackets, so only the $3$ is squared and the minus survives untouched.

Worked examples

Common mistakes

Practice problems