N8NumberFoundation & Higher

Indices & Roots

An index (or power) tells you how many times to multiply a number by itself. The laws of indices are simple shortcuts that let you combine powers without writing them out in full.

35 min Video by Maths Genie AQA GCSE Maths

Indices (Laws of Indices)Watch the walkthrough, then read the notes below.

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What you'll learn

Use powers and roots
Apply the laws of indices
Understand index 0 and negative indices
Work with fractional indices (Higher)

The laws of indices

When multiplying powers of the same base, add the indices. When dividing, subtract them. To raise a power to a power, multiply the indices.

a^{m} \times a^{n} = a^{m + n}

Multiplying: add the powers.

a^{m} \div a^{n} = a^{m - n}

Dividing: subtract the powers.

(a^{m})^{n} = a^{mn}

Power of a power: multiply.

Special indices

Anything to the power 0 is 1. A negative power means “one over”: $a^{- n} = \frac{1}{a ^{n}}$ . A fractional power is a root: $a^{1/2} = a$ (Higher).

1Same base, so add the powers.

5 + 3 = 8

Answer

x^{8}

Tip — The laws only work when the base is the same. $2^{3} \times 5^{2}$ cannot be combined.

Remember these

a^{0} = 1

Anything to the power 0.

a^{- n} = \frac{1}{a ^{n}}

Negative power.

a^{1/2} = a

Fractional power = root (Higher).

Watch out for these

Multiplying the powers when multiplying terms:

x^{5} \times x^{3} = x^{15}

Add the powers:

x^{8}

Thinking

a^{0} = 0

Anything (except 0) to the power 0 is 1.

Key takeaways

Multiply → add powers; divide → subtract powers.
Power of a power → multiply the indices.
a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^{1/2} = √a.

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