GCSE specification fit
A core GCSE Number skill that later appears in Algebra.
Powers and indices help you write repeated multiplication neatly. The same rules are used in standard form, prime factorisation, algebra and surds.
What you will learn
Why this matters
Index notation saves space and helps you see structure. Instead of writing a long string of repeated multiplication, you can write one clear power.
These laws also make later topics easier, especially standard form, algebraic expressions, compound growth and exact number work.
Prior knowledge
You should already be comfortable with:
Clear explanation
An index tells you how many times to use a number as a factor. In 2⁵, the base is 2 and the index is 5.
2⁵ = 2 × 2 × 2 × 2 × 2 = 32The most important rule is: only use the simple index laws when the base is the same.
Multiplying powers with the same base
When you multiply powers with the same base, add the indices.
aᵐ × aⁿ = aᵐ⁺ⁿDividing powers with the same base
When you divide powers with the same base, subtract the indices.
aᵐ ÷ aⁿ = aᵐ⁻ⁿPowers of powers
When a power is raised to another power, multiply the indices.
(aᵐ)ⁿ = aᵐⁿ (2³)² = 2⁶ = 64Zero and negative indices
A non-zero number to the power 0 equals 1. A negative index means a reciprocal.
7⁰ = 1 4⁻¹ = 14The phrase non-zero matters: 0⁰ is not treated as a normal GCSE calculation.
Fractional indices
Higher-tier questions may write roots as fractional powers. The denominator of the fraction tells you the root.
161/2 = √16 = 4 271/3 = ∛27 = 3Worked examples
Example 1: Simplify 2³ × 2⁴.
The base is the same, so add the indices.
2³ × 2⁴ = 2⁷Example 2: Simplify 10⁶ ÷ 10².
The base is the same, so subtract the indices.
10⁶ ÷ 10² = 10⁴Example 3: Simplify (3²)³.
A power raised to a power means multiply the indices.
(3²)³ = 3⁶Example 4: Use a fractional index
Evaluate 641/3.
641/3 = ∛64 4 × 4 × 4 = 64Quick checks
Choose an answer, then check your thinking.
1. What is 4² × 4³?
2. What is 6⁷ ÷ 6²?
3. What value does the zero index give for 9⁰?
Practice questions
Question 1
A calculator check shows two powers with the same base: 7² × 7⁵. Simplify the expression using an index law.
Reveal answer and marking guidance
Answer: 7⁷.
Marking: The base is the same, so add the indices: 2 + 5 = 7.
Question 2
A pupil writes 8⁹ ÷ 8⁴ as one power of 8. What should the simplified expression be?
Reveal answer and marking guidance
Answer: 8⁵.
Marking: The base is the same, so subtract the indices: 9 − 4 = 5.
Question 3
The expression (5³)² means a power has been raised to another power. Simplify it as a single power of 5.
Reveal answer and marking guidance
Answer: 5⁶.
Marking: A power raised to another power means multiply the indices: 3 × 2 = 6.
Question 4
Write 2⁻³ as an exact fraction, showing that a negative index means a reciprocal rather than a negative answer.
Reveal answer and marking guidance
Answer: 18.
Marking: 2⁻³ = 12³ = 18.
Question 5
A classmate says 3² × 4² = 12⁴ because the bases can be multiplied and the indices added. Explain why this is not a valid index-law step.
Reveal answer and marking guidance
Answer: The bases are different, so you cannot add the indices in that way.
Marking: 3² × 4² = 9 × 16 = 144, while 12⁴ is much larger. The simple multiplication index law needs the same base.
Question 6
Evaluate 811/2 and 1251/3.
Reveal answer and marking guidance
Answer: 811/2 = 9 and 1251/3 = 5.
Marking: A power of 12 means square root, and a power of 13 means cube root.
Question 7
Simplify 3⁴ × 9² as a power of 3.
Reveal answer and marking guidance
Answer: 3⁸.
Marking: First rewrite 9² as (3²)² = 3⁴. Then 3⁴ × 3⁴ = 3⁸.
Question 8
Evaluate 10³ × 10⁻¹.
Reveal answer and marking guidance
Answer: 10² = 100.
Marking: The base is the same, so add the indices: 3 + (−1) = 2.
Question 9
Simplify 16 × 2⁻³.
Reveal answer and marking guidance
Answer: 2.
Marking: Rewrite 16 as 2⁴. Then 2⁴ × 2⁻³ = 2¹ = 2. You can also use 2⁻³ = 18, then 16 ÷ 8 = 2.
Question 10
Simplify 27 × 3⁻²3⁴ as a fraction in its simplest form.
Reveal answer and marking guidance
Answer: 127.
Marking: Rewrite 27 as 3³. Then the numerator is 3³ × 3⁻² = 3¹, so 3¹3⁴ = 3⁻³ = 127. Check the subtraction carefully: 3 + (−2) − 4 = −3.
Answers and marking guidance
The exact practice answers are hidden under each question so you can try first. For powers and indices, marks usually come from choosing the correct index law and applying it only when the bases match: add indices when multiplying, subtract them when dividing, multiply indices for a power of a power, use reciprocals for negative indices, and translate fractional indices into roots when they are in your tier. When bases do not match at first, look for a rewrite such as 8 = 2³ or 9 = 3² before using an index law.
Common mistakes
- Multiplying the indices when multiplying powers: for 2³ × 2⁴, add the indices to get 2⁷.
- Using index laws with different bases: 3² × 4² does not become 12⁴.
- Thinking a zero index makes zero: 6⁰ = 1, not 0.
- Forgetting that negative indices mean reciprocals: 5⁻¹ = 15.
Extension challenge
Simplify this expression as far as possible:
2³ × 2⁻¹ × (2²)³Reveal answer
Answer: 2⁸ = 256.
First use (2²)³ = 2⁶. Then add the indices: 3 + (−1) + 6 = 8.
Exam-board guidance
Powers and indices are core GCSE Maths skills. The exact paper style and tier detail can vary, but every board expects you to notice matching bases, choose the right law and write negative or fractional indices carefully when they are in your tier.
AQA GCSE Maths
Index laws support number work, standard form and algebra, so always check the base before adding, subtracting or multiplying indices, and use reciprocals for negative powers.
OCR GCSE Maths
This lesson fits the Indices and Surds content. Foundation work starts with positive powers; Higher work extends to negative and fractional indices, including roots written as powers.
Pearson Edexcel GCSE Maths
Index notation appears in Number and Algebra, especially standard form, simplifying expressions and questions where different bases must not be combined.
Eduqas GCSE Maths
Use index laws for clear number working and later algebra simplification, and write one step that shows whether you are adding, subtracting or multiplying indices.
WJEC Wales
Index laws are part of the Number section and connect strongly to standard form, calculator work, powers of 10 and later algebraic manipulation.
CCEA GCSE Maths
Learn the core index laws first, then follow your unit and tier route for negative and fractional powers, especially where calculator access changes.
Next lesson
The next lesson is Standard Form, where powers of 10 are used to write very large and very small numbers neatly.