Multiply, divide and raise powers using the index laws, including the zero and negative powers. It is on Foundation and Higher, aimed at grade 4. Free questions with answers are on this page. Worked examples show each line before you try the questions.
The index laws are how powers combine. am×an=am+n, am÷an=am−n, (am)n=amn. The bases must match. 23×24=27, not 47 and not 212. A power of 0 is 1 (for a=0): 50=1. A negative power is a reciprocal: a−n=an1.
Numbers that are not powers of the same base need rewriting first: 43=(22)3=26. Brackets matter: −32=−9 but (−3)2=9. Write the law you used — it is a method mark on a 'simplify' question.
When the bases match, multiply → add the powers, divide → subtract, a power of a power → multiply the indices. Negative powers are reciprocals: 4−2=161. The number in front of a bracket is raised as well: (2x3)2=4x6.
Objectives
2⁴ = 2 × 2 × 2 × 2 = 16.
a^m × a^n = a^(m+n). The base stays.
a^m ÷ a^n = a^(m−n).
Key points
Add the powers when multiplying terms with the same base.
Subtract the powers when dividing terms with the same base.
Multiply the powers when raising a power to another power.
Know that anything to the power 0 is 1 and a negative power means 'one over'.
Worked examples
Example 1
Work out 2^2 × 2^2.
2 marks
The bases are the same, so add the powers: 2 + 2 = 4.
2^4 = 16.
The result is 16.
Answer: 16
Common slips
Slip: The bases are the same, so subtract the powers: 2 − 2 = 0. A product of powers adds the powers. Subtracting is for division.
Slip: 2^0 = 1. This follows from the wrong power.
Slip: The result is 1. The answer is far too small for a product.
Example 2
Work out 2^3 ÷ 2^2.
2 marks
The bases are the same, so subtract the powers: 3 − 2 = 1.
2^1 = 2.
The result is 2.
Answer: 2
Common slips
Slip: The bases are the same, so add the powers: 3 + 2 = 5. A quotient of powers subtracts the powers. Adding is for multiplication.
Slip: 2^5 = 32. This follows from the wrong power.
Slip: The result is 32. Dividing should make the number smaller, but this answer is bigger than the start.
Example 3
Work out (2^2)^2.
2 marks
A power of a power multiplies the powers: 2 × 2 = 4.
2^4 = 16.
The result is 16.
Answer: 16
Common slips
Slip: Multiply the base by both powers: 2 × 2 × 2 = 8. The powers are multiplied, but the base stays as 2.
Slip: The value is 8. The base should be raised to the new power, not multiplied by it.
Slip: The result is 8. The answer is far too small.
Questions and answers
GCSE Maths indices questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Simplify 2^3 × 2^4. What is the value?
Show answer
Answer: 128
Add the indices.
Then evaluate the power.
M1 — Add the indices.
A1 — Answer 128
Question 2 Easy2 marks
Simplify 3^6 ÷ 3^2. What is the value?
Show answer
Answer: 81
Subtract the indices.
The value is 81.
M1 — Subtract the indices.
A1 — Answer 81
Question 3 Medium2 marks
Write 2 to the power −2 as a fraction.
Show answer
Answer: 1/4
The positive power 2^2 is the denominator.
A negative index is the reciprocal.
M1 — The positive power 2^2 is the denominator.
A1 — Answer 1/4
Question 4 Medium2 marks
Work out 10^2.
Show answer
Answer: 100
Multiply the number by itself.
The value is 100.
M1 — Multiply the number by itself.
A1 — Answer 100
Question 5 Hard2 marks
Work out 2^3 × 2^2 ÷ 2^2. What is the value?
Show answer
Answer: 8
Add the first two indices, then subtract 2.
The single power is 2^3.
M1 — Add the first two indices, then subtract 2.
A1 — Answer 8
Question 6 Hard2 marks
Which is larger, 2^5 or 4 × 2^2? Give the larger value.
Show answer
Answer: 32
4 × 2^2 = 2^4.
2^5 is the larger value.
M1 — 4 × 2^2 = 2^4.
A1 — Answer 32
Question 7 Exam-style2 marks
Which is larger, 2^6 or 4 × 2^3? Give the larger value.
Show answer
Answer: 64
4 × 2^3 = 2^5.
2^6 is the larger value.
M1 — 4 × 2^3 = 2^5.
A1 — Answer 64
Common mistakes
Most lost marks on this topic are one slip.
Multiplying the powers when the bases are multiplied.
Add the powers.
Adding the bases.
The base stays the same.
Saying a number to the power 0 is 0.
It is 1, as long as the base is not 0.
Spot the slip
2³ means
2 × 2 × 2
2 × 3
2 + 2 + 2
Show the slip
2 × 2 × 2 The power says how many times the base is a factor.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write the law you are using as a step, e.g. x7−2, before writing x5 — the method mark is often for showing that.
On non-calculator papers, work out small powers by hand: 25=32, 34=81, 53=125 are worth knowing.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.