Indices

Description

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 $x = 3$.

Links

Manipulating algebraic fractions Fractional & negative indices Basic indices