Find \(5^2\).
Square means multiply by itself.
\(5^2 = 5 × 5 = 25\).
Powers and Roots
Powers describe repeated multiplication, while roots reverse this process. They are closely linked to simplifying expressions and surds, and understanding these concepts is important for solving problems in GCSE Maths.
Overview
Powers and roots are opposite ideas.
A power tells you how many times a number is multiplied by itself.
A root works backwards and asks which number creates the result.
\( 3^2 = 9 \qquad \sqrt{9} = 3 \)
In GCSE Maths, you need to recognise squares, cubes, square roots and cube roots, and use them accurately in number and algebra questions.
What you should understand after this topic
- Understand what powers mean
- Understand what square roots and cube roots mean
- Understand how powers and roots are connected
- Evaluate simple powers and roots
- Understand how this topic links to indices and surds
Key Definitions
Power
A short way of writing repeated multiplication.
Exponent / Index
The small number showing how many times to multiply.
Base
The main number being multiplied by itself.
Square
A number multiplied by itself once, such as \(5^2\).
Cube
A number multiplied by itself twice, such as \(4^3\).
Root
The opposite of a power. It asks which number creates the result.
Key Rules
Square
\( a^2 = a \times a \)
Cube
\( a^3 = a \times a \times a \)
Square root
\( \sqrt{a} \) means the number that squares to make \( a \).
Cube root
\( \sqrt[3]{a} \) means the number that cubes to make \( a \).
Quick Values to Know
| Number | Square | Cube |
|---|---|---|
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 5 | 25 | 125 |
| 10 | 100 | 1000 |
How to Solve
Step 1: Understand powers
A power is repeated multiplication written in a shorter form.
\( 2^4 = 2 \times 2 \times 2 \times 2 = 16 \)
The base is 2 and the exponent is 4.
Exam tip: The exponent tells you how many times the base is multiplied.
Step 2: Know squares and cubes
Learn common square numbers and cube numbers to answer quickly.
Square
\(6^2 = 6 \times 6 = 36\)
Cube
\(4^3 = 4 \times 4 \times 4 = 64\)
Step 3: Understand roots
A root is the inverse of a power. It works backwards.
\( \sqrt{49} = 7 \)
\( \sqrt[3]{27} = 3 \)
\(\sqrt{49}=7\) means 7 squared gives 49.
\(\sqrt[3]{27}=3\) means 3 cubed gives 27.
Step 4: Powers and roots undo each other
\( \sqrt{7^2} = 7 \)
\( (\sqrt{36})^2 = 36 \)
Exam thinking: If a power and matching root appear together, they often cancel.
Step 5: Be careful with negative numbers
Brackets change the meaning.
See order of operations for priority rules.
\((-3)^2\)
\(9\)
\(-3^2\)
\(-9\)
Step 6: Estimate roots
If a root is not exact, compare it with nearby square or cube numbers.
\( \sqrt{20} \)
\(16 < 20 < 25\)
So \(4 < \sqrt{20} < 5\).
Step 7: Exam method summary
- Identify whether the question uses a power or a root.
- Use known square and cube facts where possible.
- Check brackets carefully with negative numbers.
- Estimate between known values if the root is not exact.
- Round only when the question asks for it.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Evaluate \( 3^4 \).
Edexcel
Evaluate \( 10^5 \).
Edexcel
Find the value of \( \sqrt{144} \).
Edexcel
Find the value of \( \sqrt[3]{64} \).
Edexcel
Write \( 2^3 \times 2^4 \) as a single power of 2.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on index laws and accurate simplification.
AQA
Simplify \( a^5 \times a^3 \).
AQA
Simplify \( \frac{x^7}{x^2} \).
AQA
Simplify \( (m^3)^4 \).
AQA
Simplify \( y^6 \div y^2 \).
AQA
A student says that \( 2^3 + 2^3 = 2^6 \).
Tick one box. Yes ☐ No ☐
Give a reason for your answer.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, fractional indices, and problem-solving.
OCR
Write \( \sqrt{49} \) as a power of 7.
OCR
Write \( \sqrt[3]{x} \) in index form.
OCR
Simplify \( 27^{\frac{2}{3}} \).
OCR
Simplify \( 16^{\frac{3}{4}} \).
OCR
Express \( \frac{1}{x^3} \) using a negative index.
Exam Checklist
Step 1
Check whether the question is asking for a power or a root.
Step 2
Expand carefully if needed to understand the power.
Step 3
Use known square and cube numbers where possible.
Step 4
Watch brackets closely with negative numbers.
Most common exam mistakes
Notation
Reading \(a^2\) as \(a \times 2\) instead of \(a \times a\).
Roots
Giving the wrong integer because square numbers were not recognised.
Negative signs
Ignoring the difference between \((-3)^2\) and \(-3^2\).
Calculator rounding
Rounding too early or giving a decimal when an exact answer is possible.
Common Mistakes
These are common mistakes students make when working with powers and roots in GCSE Maths.
Mixing up square and cube
Incorrect
A student confuses squaring with cubing.
Correct
Squaring means multiplying a number by itself once (\(a^2 = a \times a\)), while cubing means multiplying it twice (\(a^3 = a \times a \times a\)).
Misunderstanding index notation
Incorrect
A student thinks \(a^2\) means \(a \times 2\).
Correct
An index tells you how many times to multiply the base by itself. \(a^2\) means \(a \times a\), not \(a \times 2\).
Not recognising roots as inverse operations
Incorrect
A student treats roots as unrelated to powers.
Correct
Roots undo powers. For example, \(\sqrt{a^2} = a\) (for positive values), so roots are the inverse of powers.
Ignoring brackets with negative numbers
Incorrect
A student calculates \(-3^2\) as 9 instead of \(-9\).
Correct
Without brackets, the square applies only to the number: \(-3^2 = -9\). To square the negative fully, write \((-3)^2 = 9\).
Confusing exact and approximate answers
Incorrect
A student gives a decimal when an exact root is required, or vice versa.
Correct
Leave answers in exact form (e.g. \(\sqrt{2}\)) unless the question asks for a decimal approximation.
Try It Yourself
Practise calculating powers, squares, cubes and roots.
Foundation
Foundation Practice
Work with squares, cubes and basic roots.
Find \(6^2\).
6 × 6.
\(6^2 = 36\).
Find \(3^3\).
Cube means multiply 3 times.
\(3 × 3 × 3 = 27\).
Find \(4^3\).
4 × 4 × 4.
\(4^3 = 64\).
Find \(\sqrt{49}\).
Which number squares to 49?
\(7^2 = 49\), so answer is 7.
Find \(\sqrt{81}\).
Which number squared gives 81?
\(9^2 = 81\).
Find \(\sqrt[3]{27}\).
Which number cubed gives 27?
\(3^3 = 27\).
Find \(\sqrt[3]{64}\).
Think of cubes.
\(4^3 = 64\).
Find \((-3)^2\).
Square the whole number.
\((-3)^2 = 9\).
Find \((-5)^2\).
Negative × negative = positive.
\((-5)^2 = 25\).
Higher
Higher Practice
Work with powers, negatives and index rules.
Find \(2^4\).
Multiply 2 four times.
\(2^4 = 16\).
Find \(3^4\).
3 × 3 × 3 × 3.
\(3^4 = 81\).
Find \(-3^2\).
Power first, then negative.
\(-3^2 = -(3^2) = -9\).
Find \((-2)^3\).
Multiply 3 negatives.
\((-2)^3 = -8\).
Find \(\sqrt{144}\).
Which number squared gives 144?
\(12^2 = 144\).
Find \(\sqrt{121}\).
Square numbers.
\(11^2 = 121\).
Find \(\sqrt[3]{125}\).
Cube numbers.
\(5^3 = 125\).
Find \(\sqrt[3]{216}\).
Think cubes.
\(6^3 = 216\).
A student says \((-4)^2 = -16\). What is wrong?
Negative × negative = positive.
\((-4)^2 = 16\), not -16.
Find \(10^3\).
Multiply by 10 three times.
\(10^3 = 1000\).
Games
Practise this topic with interactive games.
Games coming soon.
Powers and Roots Video Tutorial
Frequently Asked Questions
What does a power mean?
A power shows how many times a number is multiplied by itself.
What is a square root?
A square root is a number that multiplies by itself to give the original number.
How do I simplify powers?
Use index laws such as adding powers when multiplying and subtracting when dividing.