Laws of indices (exponents)

Multiply, divide and raise powers; work with zero, negative and fractional indices.

Also known as: exponent rules, powers, negative indices, fractional indices.

SAT · Advanced Math GCSE Maths · Number IGCSE Maths · Number

Key ideas

$a^m \times a^n = a^{m+n}$ and $a^m \div a^n = a^{m-n}$ .
$(a^m)^n = a^{mn}$ and $(ab)^n = a^n b^n$ .
$a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$ (for $a \neq 0$ ).
$a^{m/n} = \left(\sqrt[n]{a}\right)^m$ .

The core laws

Powers with the same base combine by adding or subtracting indices: $x^4 \times x^3 = x^7, \qquad \frac{x^9}{x^2} = x^7.$ A power of a power multiplies: $(x^3)^4 = x^{12}$ . Every factor inside a bracket is raised: $(3a^2b)^3 = 27a^6b^3$ .

Zero and negative indices

Any non-zero number to the power $0$ equals $1$ . A negative index means a reciprocal: $4^{-1} = \frac{1}{4}, \qquad 2^{-3} = \frac{1}{8}, \qquad \left(\frac{2}{3}\right)^{-2} = \frac{9}{4}.$

Fractional indices and equations

The denominator is a root and the numerator is a power: $27^{2/3} = (\sqrt[3]{27})^2 = 9$ . Take the root first to keep numbers small.

To solve $2^{x+3} = 8^{x-1}$ , write both sides with the same base: $8 = 2^3$ , so $2^{x+3} = 2^{3x-3}$ and $x + 3 = 3x - 3$ , giving $x = 3$ .

Worked examples

Example 1 · Fractional and negative index

Evaluate $\left(\frac{16}{81}\right)^{-3/4}$ .

Step 1.Negative index: flip the fraction → $\left(\frac{81}{16}\right)^{3/4}$ .
Step 2.Fourth root: $\frac{3}{2}$ .
Step 3.Cube: $\frac{27}{8}$ .

Answer: $\frac{27}{8}$

Example 2 · Equation with powers

Solve $2^{x+3} = 8^{x-1}$ .

Step 1. $8^{x-1} = (2^3)^{x-1} = 2^{3x-3}$ .
Step 2.Equate indices: $x + 3 = 3x - 3$ .
Step 3. $x = 3$ .

Answer: $x = 3$

Common mistakes

Multiplying indices when multiplying powers: $x^4 \times x^3 \neq x^{12}$ .
Thinking a negative index makes the number negative: $2^{-3} = \frac{1}{8}$ , not $-8$ .
Forgetting to raise the coefficient: $(3a^2)^3 = 27a^6$ , not $3a^6$ .

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Laws of indices (exponents)Easier

Which expression is equivalent to $x^4 \cdot x^3$ ?

Question 2Laws of indices (exponents)Easier

Evaluate $5^0 + 4^{-1}$ .