You have written powers of 10 like $10^3 = 10 \times 10 \times 10 = 1,000$. The same short way of writing works for any number.
Key idea
$b^n$ means $b$ used as a factor $n$ times. The base $b$ is the number multiplied; the exponent $n$ counts the factors: $2^4 = 2 \times 2 \times 2 \times 2 = 16$.
Why it works
Multiplication is a short way to write repeated addition: $4 + 4 + 4 = 3 \times 4$. An exponent is a short way to write repeated multiplication: $4 \times 4 \times 4 = 4^3$. That is why the exponent counts how many times the base appears, not something to multiply by. Squares and cubes get their names from shapes: $4^2$ is the number of unit squares in a 4 by 4 square, and $3^3$ is the number of unit cubes in a 3 by 3 by 3 cube.
Worked example: Evaluate a power
Find $5^3$.
- Write 5 as a factor 3 times: $5 \times 5 \times 5$.
- $5 \times 5 = 25$, and $25 \times 5 = 125$.
Answer: $125$
Worked example: Fractions and decimals as bases
Find $\left(\frac{1}{2}\right)^3$ and $0.3^2$.
- $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$.
- $0.3 \times 0.3 = 0.09$ (3 tenths of 3 tenths is 9 hundredths).
- A base less than 1 gets smaller as the exponent grows.
Answer: $\frac{1}{8}$ and $0.09$
| Power | $2^1$ | $2^2$ | $2^3$ | $2^4$ | $2^5$ | $2^6$ |
|---|---|---|---|---|---|---|
| Value | 2 | 4 | 8 | 16 | 32 | 64 |
Common mistake
Multiplying the base by the exponent: $2^4$ is not $2 \times 4 = 8$. Quick check: write out the factors. $2^4 = 2 \times 2 \times 2 \times 2$, which is 16. The table shows the difference grows fast: $2^6 = 64$, but $2 \times 6$ is only 12.