Exponent

The two parts of a power

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

Part	Name	What it does
$2$	base	the number being multiplied
$5$	exponent	how many of them there are

The whole expression $2^5$ is called a power, so “the fifth power of two” and “two to the exponent five” name the same thing.

What the exponent is not

It is not something to multiply by. $2^5$ is $32$, while $2 \times 5$ is $10$ — and the gap widens with every step, since $2^{10}$ is $1024$ against $2 \times 10 = 20$.

See what exponents are for why this is the error worth guarding against first.

Exponents that are not positive whole numbers

The counting picture runs out at zero, but the pattern does not. Each step down divides by the base, which carries straight on past $1$:

$$2^2 = 4 \qquad 2^1 = 2 \qquad 2^0 = 1 \qquad 2^{-1} = \tfrac{1}{2}$$

So a zero exponent gives $1$ and a negative exponent gives a reciprocal. Neither is a special rule — both are the same ladder continued.

How exponents are said aloud

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

“Squared” and “cubed” come from geometry: $s^2$ is the area of a square with side $s$, and $s^3$ is the volume of a cube with edge $s$.