Indices (also called powers or exponents) follow a set of rules that let you simplify expressions without a calculator.
The index laws
| Law | Example |
|---|---|
| $a^m \times a^n = a^{m+n}$ | $x^3 \times x^4 = x^7$ |
| $a^m \div a^n = a^{m-n}$ | $x^5 \div x^2 = x^3$ |
| $(a^m)^n = a^{mn}$ | $(x^3)^2 = x^6$ |
| $a^0 = 1$ | $7^0 = 1$ |
| $a^1 = a$ | $5^1 = 5$ |
Negative indices
A negative index means "one over". Think of it as the index going below zero and flipping to the denominator:
$$a^{-n} = \frac{1}{a^n}$$
Examples:
- $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
- $x^{-1} = \frac{1}{x}$
- $5^{-2} = \frac{1}{25}$
Change of base
Sometimes a number can be rewritten as a power of a simpler base. This is useful when solving equations where both sides need to share the same base.
Examples:
- $8 = 2^3$, so $8^x = (2^3)^x = 2^{3x}$
- $\frac{1}{9} = 3^{-2}$
- $\sqrt{5} = 5^{\frac{1}{2}}$
If a question gives you something like $2^x = 8$, rewrite 8 as $2^3$ so both sides have the same base, giving $x = 3$.