The laws of indices come up on every single GCSE Maths paper — foundation and higher — and they don't take long to master. AQA (8300), Edexcel (1MA1), and OCR (J560) all include index questions, typically in the first half of non-calculator Paper 1 and then again in the later algebraic sections of Papers 2 and 3. A solid understanding of index laws can earn you quick, clean marks and also unlocks other topics like surds, standard form, and exponential graphs.
What You Need to Know First
Check you're comfortable with these before working through the index laws:
- What a base and power (or index/exponent) mean — in 3⁴, the base is 3 and the power is 4
- That 3⁴ = 3 × 3 × 3 × 3 = 81 (the base is multiplied by itself that many times)
- Square roots and cube roots — you'll need these when working with fractional indices
- Multiplying and dividing fractions — required for evaluating fractional powers
- Basic negative numbers — needed for understanding negative indices
The Core Index Laws
There are five index laws you must know. Learn them in this order — the first three form the foundation, and the last two (zero and negative indices) follow logically from them.
Law 1 — Multiplying same bases: a^m × a^n = a^(m + n)
Law 2 — Dividing same bases: a^m ÷ a^n = a^(m − n)
Law 3 — Power of a power: (a^m)^n = a^(mn)
Law 4 — Zero index: a^0 = 1 (for any a ≠ 0)
Law 5 — Negative index: a^(−n) = 1 / a^n
Worked Example 1:
Simplify: 5x³ × 3x⁴
The number parts multiply normally: 5 × 3 = 15
The x parts use Law 1: x³ × x⁴ = x^(3 + 4) = x⁷
Answer: 15x⁷
Worked Example 2:
Simplify: (2x²)³
Apply Law 3 — raise both the number and the variable to the power of 3:
2³ = 8
(x²)³ = x^(2 × 3) = x⁶
Answer: 8x⁶
Students often forget to cube the number in front. Bracket notation like (2x²)³ means everything inside is raised to that power.
Worked Example 3:
Simplify: 12a⁵b³ ÷ 4a²b
Divide the numbers: 12 ÷ 4 = 3
Apply Law 2 to each variable: a⁵ ÷ a² = a^(5 − 2) = a³ b³ ÷ b¹ = b^(3 − 1) = b²
Answer: 3a³b²
Negative and Fractional Indices
This is where most students lose marks — negative and fractional indices are tested heavily on higher tier and often appear on the final questions of Paper 1 without a calculator.
Negative Indices
A negative index means take the reciprocal (flip it). That's all.
$$a^{-n} = \frac{1}{a^n}$$
So 2^(−3) = 1/2³ = 1/8.
Worked Example 4:
Evaluate: 5^(−2)
$$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$$
Answer: 1/25
When the base is a fraction, flip the fraction and make the power positive:
$$\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4}$$
Fractional Indices
A fractional index links powers to roots. The denominator of the fraction tells you which root to take:
$$a^{\frac{1}{n}} = \sqrt[n]{a}$$
So 8^(1/3) = ∛8 = 2.
When the numerator isn't 1, the index combines a root and a power:
$$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$$
The order doesn't matter mathematically, but always take the root first — it keeps the numbers smaller and avoids huge arithmetic.
Worked Example 5:
Evaluate: 8^(2/3)
Step 1: Take the cube root first (denominator = 3): ∛8 = 2
Step 2: Raise to the power 2 (numerator = 2): 2² = 4
Answer: 4
Worked Example 6:
Evaluate: 27^(−2/3)
This combines a fractional index and a negative index.
Step 1: Deal with the negative — take the reciprocal: 27^(−2/3) = 1 / 27^(2/3)
Step 2: Evaluate 27^(2/3): cube root of 27 = 3, then 3² = 9
Step 3: Take the reciprocal: 1/9
Answer: 1/9
That three-step approach — reciprocal, root, power — handles all negative fractional index questions reliably.
Worked Example 7:
Simplify: x^(3/4) × x^(1/4)
Apply Law 1: x^(3/4 + 1/4) = x^(4/4) = x^1 = x
Answer: x
Common Mistakes Students Make
- Applying the negative index to the wrong part. 4^(−3) is NOT −64. The negative makes it a reciprocal: 1/64. The answer is always positive when the base is positive.
- Taking the power before the root for fractional indices. Evaluating 64^(2/3) by doing 64² = 4096 first, then ∛4096, is correct but unnecessarily hard. Take the root first: ∛64 = 4, then 4² = 16. Much simpler.
- Forgetting that a^0 = 1 for any non-zero base. This includes expressions like (5x + 3)^0 = 1. Students sometimes write 0.
- Only applying the power to the variable, not the coefficient. In (3x)² = 9x², students often write 3x² — the coefficient must be squared too.
- Misreading fractional index questions on Paper 1. Without a calculator, the only numbers that work neatly with fractional indices are perfect squares, cubes, or fourth powers. If you're getting a messy decimal, re-read the question.
Exam Tips for Indices
- Learn the five laws cold — AQA, Edexcel, and OCR all reward marks for applying them correctly without prompting. You won't be given the laws in an exam.
- On non-calculator papers, look out for base numbers that are powers of something simple: 8 = 2³, 27 = 3³, 16 = 2⁴, 125 = 5³. The question is almost always designed so the root gives a whole number.
- Edexcel 1MA1 regularly asks you to simplify algebraic expressions with indices — think 3x²y × 4x³y². Apply each law term by term: numbers first, then each variable separately.
- When a question says "without a calculator, evaluate" and gives you a fractional or negative index — that is a strong signal to spot the root and apply it first.
- Show each law as a separate line of working. OCR and AQA mark schemes typically award one method mark per law applied, so rushing to a final answer without showing steps costs marks.
Practise Now
The best way to get confident with the laws of indices is to practise exam-style questions with instant feedback. Try indices questions now on GCSEMathsAI — AI-generated questions matched to your board and tier, with detailed marking and personalised feedback.