Laws of Indices — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the rules for working with powers. By the end of this guide you should be able to multiply, divide and raise powers to further powers, and handle a power of zero.

You should also be able to work with negative indices, fractional indices including roots, and combinations of the two, as well as simplify expressions containing coefficients alongside their powers.

The organising idea is that an index is just a shorthand for repeated multiplication, and every rule is that shorthand being counted. Writing x³ × x² out in full gives x × x × x × x × x — five x's, so x⁵. Nobody needs to memorise "add the indices"; it is simply what counting the x's produces. The same goes for dividing, where the cancelling leaves a shorter run, and for a power of a power, where you get so many groups of so many. If a rule ever slips, write the powers out in full for small numbers and read the answer off. That also explains the cases that seem strange — a power of zero, and negative and fractional indices — which are not arbitrary but exactly what keeps the counting consistent.

Key terms and definitions

Index (plural indices) — the small raised number showing how many times the base is multiplied by itself.

Base — the number or letter being raised to a power. In x⁵, the base is x.

Power — another word for an index, and also used for the whole expression.

Coefficient — the ordinary number in front, as the 3 in 3x².

Reciprocal — 1 divided by a quantity.

Root — the inverse of a power. The square root undoes squaring.

Core concepts

Multiplying: add the indices

When the bases are the same, multiplying means adding the indices:

aᵐ × aⁿ = aᵐ⁺ⁿ

So x³ × x⁴ = x⁷, which you can check by counting: three x's followed by four x's is seven x's.

The rule applies only when the bases match. There is no way to simplify x³ × y², because the two letters count separately.

Dividing: subtract the indices

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

So x⁵ ÷ x² = x³. Writing it as a fraction shows why: five x's on top and two underneath cancel in pairs, leaving three.

A power of a power: multiply the indices

(aᵐ)ⁿ = aᵐⁿ

So (x²)³ = x⁶, because you have three groups of two x's.

When a power applies to a bracket containing more than the base, it reaches everything inside. So (3x²)³ means 3³ × (x²)³ = 27x⁶ — the coefficient is cubed as well, and forgetting that is one of the commonest errors in the topic.

The power of zero

a⁰ = 1 for any non-zero a.

This is not an arbitrary convention. Consider x³ ÷ x³: the answer is plainly 1, since anything divided by itself is 1. But the subtraction rule gives x⁰. Both must be right, so x⁰ = 1.

Note that the coefficient is not swallowed: 5x⁰ means 5 × 1 = 5, whereas (5x)⁰ = 1.

Negative indices: take the reciprocal

a⁻ⁿ = 1 ÷ aⁿ

So x⁻² = 1/x², and 2⁻³ = 1/8.

Again the subtraction rule forces it. x² ÷ x⁵ gives x⁻³ by the rule, but cancelling directly leaves 1/x³. So x⁻³ and 1/x³ are the same thing.

A negative index therefore means "one over", not a negative answer. The value of 2⁻³ is 1/8, which is positive and small.

With a fraction as the base, a negative index flips it: (2/3)⁻¹ = 3/2.

Fractional indices: roots

A unit fraction index means a root:

a^(1/2) = √a, and a^(1/3) is the cube root.

The multiplying rule explains it. Since a^(1/2) × a^(1/2) = a¹ = a, the quantity a^(1/2) must be whatever multiplies by itself to give a — which is the square root.

So 9^(1/2) = 3, and 8^(1/3) = 2.

For a general fraction, the bottom is the root and the top is the power:

a^(m/n) = the nth root of a, then raised to the power m

So 8^(2/3) means take the cube root of 8 to get 2, then square it to get 4.

Take the root first. Doing the power first gives 8² = 64 and then the cube root of 64 = 4, which happens to agree here but produces far larger numbers and is much harder without a calculator. Rooting first keeps everything small.

Negative fractional indices

These combine both rules: take the reciprocal, and treat the fraction as root then power.

So 16^(−1/2) = 1 ÷ 16^(1/2) = 1 ÷ 4 = 1/4.

And 27^(−2/3) = 1 ÷ 27^(2/3). The cube root of 27 is 3, squared is 9, so the answer is 1/9.

Work through it in order: deal with the negative first by writing the reciprocal, then the root, then the power.

Coefficients and indices together

Numbers in front are handled separately from the powers.

For 3x² × 4x³: multiply the coefficients to get 12, add the indices to get x⁵, so the answer is 12x⁵.

For 20x⁶ ÷ 5x²: divide the coefficients to get 4, subtract the indices to get x⁴, so the answer is 4x⁴.

Adding the coefficients while adding the indices is a frequent muddle. The coefficients follow ordinary arithmetic; only the indices obey the index laws.

Solving equations with indices

Some questions give an equation such as 2ˣ = 32 and ask for x.

Write both sides as powers of the same base. Since 32 = 2⁵, the equation becomes 2ˣ = 2⁵, so x = 5.

The step that makes it work is matching the bases. For 9ˣ = 27, rewrite both as powers of 3: 9 = 3² and 27 = 3³, so the equation becomes 3²ˣ = 3³, giving 2x = 3 and x = 1.5.

Once the bases match, the indices can simply be equated, because a power determines its value uniquely.

Indices and standard form

Standard form is an application of these laws, which is why the two topics reinforce each other.

Multiplying numbers in standard form adds the indices of 10, and dividing subtracts them, exactly as the rules above require. So (3 × 10⁵) × (2 × 10³) has index 5 + 3 = 8.

A negative index in standard form is the same negative index met here: 10⁻³ means one thousandth, since it is the reciprocal of 10³.

Recognising that standard form uses no new rules makes both topics shorter to learn.

Powers with different bases

The laws need matching bases, so sometimes a base must be rewritten first.

To simplify 2³ × 4², notice that 4 = 2², so 4² = (2²)² = 2⁴. The expression becomes 2³ × 2⁴ = 2⁷.

Spotting that one base is a power of the other is a standard higher-tier step.

Worked examples

Example 1: Combining several rules

Simplify (2x³)⁴ ÷ (4x⁵).

Deal with the bracket first, applying the power to everything inside: 2⁴ × x¹² = 16x¹².

Now divide: the coefficients give 16 ÷ 4 = 4, and the indices subtract to give x⁷.

The answer is 4x⁷.

Cubing only the x and leaving the 2 alone would have given 2x¹², a wrong answer that is easy to avoid by writing the bracket expansion as its own step.

Example 2: A negative fractional index

Work out 16^(−3/4) without a calculator.

The negative index means take the reciprocal: 1 ÷ 16^(3/4).

Now the fraction: the bottom, 4, is the root, so take the fourth root of 16, which is 2.

The top, 3, is the power, so cube it: 2³ = 8.

So 16^(3/4) = 8, and the answer is 1/8.

Doing the power first would have meant finding the fourth root of 4096, which is the same answer by a much harder route.

Example 3: Rewriting a base

Simplify 9² × 3⁵, giving the answer as a power of 3.

The bases differ, so the laws cannot be applied directly. But 9 = 3², so 9² = (3²)² = 3⁴.

Now the bases match: 3⁴ × 3⁵ = 3⁹.

Recognising that one base is a power of the other is what makes the question solvable.

Common mistakes and how to avoid them

Multiplying the indices when multiplying terms. x³ × x⁴ = x⁷, not x¹². Count the x's if unsure.

Applying a power to only part of a bracket. (3x²)³ = 27x⁶. The coefficient is cubed too.

Treating a negative index as a negative answer. 2⁻³ = 1/8, which is positive.

Doing the power before the root in a fractional index. Take the root first; the numbers stay much smaller.

Adding the coefficients as well as the indices. 3x² × 4x³ = 12x⁵ — multiply the numbers, add the powers.

Using the laws on different bases. Rewrite one base as a power of the other first.

Forgetting that a coefficient survives a zero index. 5x⁰ = 5, but (5x)⁰ = 1.

Exam technique for "Laws of Indices"

Expand any bracket with an outside power as its own written line, applying the power to the coefficient and the letters separately.

Handle coefficients and indices in two distinct steps, so neither rule gets applied to the wrong part.

For a negative fractional index, write the reciprocal first, then the root, then the power — one line each.

Check whether bases can be made to match before deciding a question cannot be simplified.

State the answer in the form the question asks for: "as a power of 3" wants 3⁹, not 19683.

If a rule slips, write the powers out in full for a small case and read off the pattern. It takes seconds and settles it.

Quick revision summary

An index is shorthand for repeated multiplication, and every law is that shorthand being counted.

Multiplying adds the indices: aᵐ × aⁿ = aᵐ⁺ⁿ. Dividing subtracts them. A power of a power multiplies them: (aᵐ)ⁿ = aᵐⁿ.

The laws need the same base. Rewrite one base as a power of the other where possible — 9² becomes 3⁴.

A power outside a bracket applies to everything inside, so (3x²)³ = 27x⁶.

a⁰ = 1, because a³ ÷ a³ must equal both 1 and a⁰.

A negative index means a reciprocal, not a negative value: x⁻² = 1/x² and 2⁻³ = 1/8.

A fractional index means a root: the bottom is the root and the top is the power, so 8^(2/3) = 4. Take the root first to keep the numbers small.

Handle coefficients separately: 3x² × 4x³ = 12x⁵.