🔢
Number · Foundation & Higher

Powers & roots

Powers (indices) tell you how many times to multiply a number by itself. Roots are the inverse operation.

🔑

Key facts to remember

  • 1a² means a × a; a³ means a × a × a.
  • 2√a is the square root of a (the positive root unless stated otherwise).
  • 3∛a is the cube root of a.
  • 4Any number to the power 0 = 1 (e.g. 5⁰ = 1).
  • 5Any number to the power 1 = itself.
  • 6Negative power: a⁻ⁿ = 1/aⁿ
📐

Formulas

Multiply same base
aᵐ × aⁿ = a^(m+n)
Divide same base
aᵐ ÷ aⁿ = a^(m−n)
Power of a power
(aᵐ)ⁿ = a^(mn)
Negative index
a⁻ⁿ = 1/aⁿ
✍️

Worked examples

Example 1

Simplify 2⁵ × 2³

Working

  1. Same base (2), so add the powers: 5 + 3 = 8
Answer2⁸ = 256
Example 2

Evaluate 4⁻²

Working

  1. 4⁻² = 1/4² = 1/16
Answer1/16
⚠️

Common mistakes

✗Multiplying the bases instead of adding the powers (e.g. 2³ × 2⁴ ≠ 4⁷).
✗Thinking a negative power gives a negative answer — it gives a fraction.
🎯

Exam tips

✓Only add/subtract powers when the bases are the same.
✓Learn the squares to 15² and cubes to 5³ — saves time in non-calculator papers.

Ready to test yourself on Powers & roots?

Get AI-marked practice questions on exactly this subtopic.

Practice this topic →
← All topicsDashboard

▶️ Watch on YouTube

Free video lessons

Click a topic to search

▶GCSE powers and indices▶laws of indices GCSE▶index laws GCSE maths▶negative fractional indices GCSE

Opens YouTube — pick any free GCSE video.