GCSE — mathsHigher2 min read

Indices and Powers

Master the laws of indices for GCSE Maths. Apply rules for multiplying, dividing, and raising powers, including negative and fractional indices.

Tutor AI Team2026-02-25T01:42:18.587Z

Indices (also called exponents or powers) are a shorthand for repeated multiplication. The laws of indices allow you to simplify complex expressions efficiently. This topic is essential for both Foundation and Higher GCSE.

Laws of Indices

Rule	Law	Example
Multiplying	$a^m \times a^n = a^{m+n}$	$2^3 \times 2^4 = 2^7$
Dividing	$a^m \div a^n = a^{m-n}$	$5^6 \div 5^2 = 5^4$
Power of a power	$(a^m)^n = a^{mn}$	$(3^2)^4 = 3^8$
Zero index	$a^0 = 1$	$7^0 = 1$
Negative index	$a^{-n} = \frac{1}{a^n}$	$2^{-3} = \frac{1}{8}$
Fractional index	$a^{\frac{1}{n}} = \sqrt[n]{a}$	$8^{\frac{1}{3}} = 2$
Combined fractional	$a^{\frac{m}{n}} = (\sqrt[n]{a})^m$	$27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 9$

Strategy: Fractional Indices

Step 1: Root first (denominator of the fraction). Step 2: Then power (numerator of the fraction).

$16^{\frac{3}{4}} = (\sqrt[4]{16})^3 = 2^3 = 8$

Worked Example: Simplify $\frac{x^5 \times x^3}{x^2}$

Problem

$= \frac{x^8}{x^2} = x^6$

Solution

Worked Example: Evaluate $125^{\frac{2}{3}}$

Problem

$= (\sqrt[3]{125})^2 = 5^2 = 25$

Solution

Worked Example: Simplify $(2x^3)^4$

Problem

$= 2^4 \times x^{12} = 16x^{12}$

Solution

Worked Example: Write $\frac{1}{x^3}$ using negative indices.

Problem

$= x^{-3}$

Solution

Common Mistakes

Adding indices when you should multiply (power of a power). $(a^2)^3 = a^6$ , not $a^5$ .
Forgetting to apply the power to coefficients. $(3x)^2 = 9x^2$ , not $3x^2$ .
Confusing negative indices with negative numbers. $2^{-3} = \frac{1}{8}$ , not $-8$ .

Practice Problems

1. Simplify $a^4 \times a^{-2}$ .
1. Evaluate $64^{\frac{1}{3}}$ .
1. Simplify $(5x^2y)^3$ .
1. Write $\frac{3}{x^2}$ using negative indices.
1. Evaluate $32^{\frac{3}{5}}$ .

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Key Takeaways

✓
Multiply: add indices. Divide: subtract indices. Power of power: multiply indices.
✓
$a^0 = 1$ always (when $a \neq 0$ ).
✓
Negative index = reciprocal. Fractional index = root.
✓
For $a^{\frac{m}{n}}$ : root first (denominator), then power (numerator).

Tags:

#gcse#maths#number#indices#powers#exponents

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Laws of Indices

Strategy: Fractional Indices

Worked Example: Simplify $\frac{x^5 \times x^3}{x^2}$

Worked Example: Evaluate $125^{\frac{2}{3}}$

Worked Example: Simplify $(2x^3)^4$

Worked Example: Write $\frac{1}{x^3}$ using negative indices.

Common Mistakes

Practice Problems

Key Takeaways

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