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A negative index means take the reciprocal, and any non-zero number to the power 0 equals 1. Free GCSE AQA notes with worked examples and 14 questions answered.
Working with Indices and Roots is part of Structure and calculation in GCSE AQA Higher Maths.
5 min
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GCSE AQA Higher
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A negative index means take the reciprocal, and any non-zero number to the power 0 equals 1. Free GCSE AQA notes with worked examples and 14 questions answered.
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Use this lesson to stay inside Structure and calculation while connecting the detail back to the wider Number topic.
Read the core explanations from Working with Indices and Roots before testing yourself from memory.
You already know that $3 \times 3 \times 3 \times 3$ is a bit tedious to write out. Indices give you a shorthand: $3^4$. That one idea unlocks an enormous amount of maths — from standard form to surds to algebraic simplification.
Roots are the reverse operation. If squaring a number gives you the result, the square root takes you back. These two tools — indices and roots — appear on both questions without a calculator and calculator-allowed exam questions , so you need fluency with both the concept and the calculation.
The index tells you the count of multiplications, not the count of numbers. So $5^3$ involves three 5s multiplied together, giving 125 — not $5 \times 3 = 15$. That confusion is one of the most common errors at GCSE, and examiners know it.
Here are the key index values you need to know for questions without a calculator. Commit these to memory — they come up repeatedly in questions without a calculator.
| Expression | Expanded form | Value |
|---|---|---|
| $2^1$ | $2$ | $2$ |
| $2^2$ | $2 \times 2$ | $4$ |
| $2^3$ | $2 \times 2 \times 2$ | $8$ |
| $2^4$ | $2 \times 2 \times 2 \times 2$ | $16$ |
| $3^2$ | $3 \times 3$ | $9$ |
| $3^3$ | $3 \times 3 \times 3$ | $27$ |
| $4^2$ | $4 \times 4$ | $16$ |
| $5^2$ | $5 \times 5$ | $25$ |
| $10^3$ | $10 \times 10 \times 10$ | $1000$ |
Two index rules catch students out more than any others: the zero index and negative indices. Both follow logically from the pattern of powers, so let's build up to them rather than just stating the rules.
Look at powers of 2 in descending order: $2^4 = 16$, $2^3 = 8$, $2^2 = 4$, $2^1 = 2$. Each time the index drops by 1, you divide by 2. Continuing that pattern: $2^0 = 2 \div 2 = 1$. Then $2^{-1} = 1 \div 2 = \frac{1}{2}$. The pattern is consistent — the rules are not arbitrary.
A negative index does not make the answer negative. It makes the answer a fraction (a reciprocal). $5^{-2}$ is not $-25$; it is $\frac{1}{25}$. This is one of the most penalised errors in GCSE mark schemes.
Worked example: Evaluating Expressions with Zero and Negative Indices
Calculate: (a) $7^0$ (b) $3^{-2}$ (c) $2^{-4}$
Answer: (a) $7^0 = 1$ (b) $3^{-2} = \frac{1}{9}$ (c) $2^{-4} = \frac{1}{16}$
Now that you can evaluate individual powers, you need to know how to combine them. Three index laws cover the vast majority of GCSE questions. Each one follows directly from what an index means — you don't need to memorise them blindly if you understand the logic.
The multiplication law works because $a^3 \times a^4$ means $(a \times a \times a) \times (a \times a \times a \times a)$, which is seven $a$s multiplied together: $a^7$. You're counting the total number of factors, so you add the indices.
The division law works by cancellation. $\frac{a^5}{a^2} = \frac{a \times a \times a \times a \times a}{a \times a}$. Two $a$s cancel from top and bottom, leaving $a^3$. Subtracting the indices gives the same result: $5 - 2 = 3$.
Worked example: Applying Index Laws to Simplify Expressions
Simplify: (a) $x^5 \times x^3$ (b) $y^8 \div y^3$ (c) $(z^2)^5$
Answer: (a) $x^8$ (b) $y^5$ (c) $z^{10}$
Roots are the inverse of powers. If $4^2 = 16$, then $\sqrt{16} = 4$. If $3^3 = 27$, then $\sqrt[3]{27} = 3$. Knowing your square numbers and cube numbers from the table earlier means you can read roots off directly — no calculator needed.
Cube roots behave differently from square roots in one important way: you can take the cube root of a negative number. $\sqrt[3]{-8} = -2$ because $(-2) \times (-2) \times (-2) = -8$. Square roots of negative numbers, however, are not real numbers and are outside GCSE scope.
Worked example: Calculating with Roots (Without a Calculator)
Calculate: (a) $\sqrt{144}$ (b) $\sqrt[3]{64}$ (c) $\sqrt[3]{-125}$
Answer: (a) $\sqrt{144} = 12$ (b) $\sqrt[3]{64} = 4$ (c) $\sqrt[3]{-125} = -5$
Exam questions often combine roots and indices in a single expression. The strategy is always the same: deal with roots and powers one step at a time, working from the innermost operation outward.
You also need to be comfortable applying index laws when the indices are negative or when the expression involves a fraction base. These appear on both calculator-allowed questions and questions without a calculator, so the method must be secure.
Worked example: Multi-Step Calculation with Indices and Roots
Work out the value of $\left(\frac{1}{4}\right)^{-2} + \sqrt[3]{27} - 2^0$
Answer: $18$
The best way to consolidate these methods is to work through a range of question types. The problems below cover all the key skills from this lesson. Attempt each one before checking the solution — that retrieval effort is what builds long-term recall.
The index and roots topic is one where a small number of errors account for the majority of lost marks. The mistakes below are predictable — which means they are avoidable once you know what to watch for.
Across this lesson, you have covered all three specification requirements: calculating with positive integer indices, applying the zero and negative index rules, and evaluating square and cube roots. These skills connect directly — the index laws work for negative indices too, and roots can be expressed as fractional indices (a topic you may meet later in the course).
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-21.
Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
A negative index means take the reciprocal, so it makes the answer a fraction, not a negative number. For example, 3⁻² = 1/3² = 1/9, and 5⁻² is 1/25, not -25.
The cube root of 27 is 3, because 3 × 3 × 3 = 27. Knowing cube numbers such as 3³ = 27 and 4³ = 64 means cube roots can be read off without a calculator.
Any non-zero number raised to the power 0 equals 1, so 7⁰ = 1 and 2⁰ = 1. This follows the pattern of powers of 2: each time the index drops by 1 the value is divided by 2, so 2⁰ = 2 ÷ 2 = 1.
The power of a power rule says that when a power is raised to another power, the indices are multiplied. For example, (z²)⁵ = z to the power 2 × 5 = z¹⁰.
A negative number does have a real cube root. The cube root of -125 is -5, because (-5) × (-5) = 25 and 25 × (-5) = -125; square roots of negative numbers, however, are not real numbers.
A negative index on a fraction means flip the fraction, then apply the positive index. For example, (1/4)⁻² = 4² = 16, so (1/4)⁻² + cube root of 27 - 2⁰ = 16 + 3 - 1 = 18.
5 to the power of -2 is 1/25, not -25. A negative index means take the reciprocal, so 5⁻² = 1 ÷ 5² = 1/25.
A negative power does not make the answer negative; it makes it a fraction, the reciprocal. For example, 3⁻² = 1/3² = 1/9 and 2⁻⁴ = 1/16, and writing 5⁻² as -25 is one of the most penalised errors.
2 to the power of -4 is 1/16. The negative index flips the power to the denominator, 2⁻⁴ = 1/2⁴, and 2⁴ = 2 × 2 × 2 × 2 = 16.
When multiplying powers of the same base, the indices are added. This works because a³ × a⁴ means seven a's multiplied together, which is a⁷, and in the same way x⁵ × x³ = x⁸.
When dividing powers of the same base, the indices are subtracted. For example, a⁵ ÷ a² leaves a³ because two a's cancel from top and bottom, and y⁸ ÷ y³ = y⁵.
49 has two square roots, 7 and -7, because 7 × 7 = 49 and (-7)² = 49 too. However, √49 refers to the positive, or principal, root, so √49 = 7.
5 cubed is not the same as 5 times 3. 5³ means three 5s multiplied together, 5 × 5 × 5 = 125, whereas 5 × 3 = 15; confusing the two is one of the most common errors with indices.
In the expression aⁿ, a is the base and n is the index, also called the power or exponent. It means multiply a by itself n times, so in 5³ the base is 5 and the index is 3, giving 125.