This topic summary covers Knowledge Organiser: Index Laws within Index Laws for GCSE Mathematics. Revise Index Laws in Number for GCSE Mathematics with 14 exam-style questions and 22 flashcards. This topic shows up very often in GCSE exams, so students should be able to explain it clearly, not just recognise the term. It is section 14 of 14 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Knowledge Organiser: Index Laws
Key Terms
- Index law: A rule for simplifying expressions involving powers
- Base: The number or letter being raised to a power
- Zero power: Any non-zero number raised to the power 0 equals 1
- Negative power: a⁻ⁿ = 1/aⁿ (reciprocal)
- Fractional index: a^(m/n) = (ⁿ√a)ᵐ (Higher tier)
Must-Know Facts
- Laws only apply when the BASES are the same
- 2³ × 3² ≠ 6⁵ — different bases cannot be combined
- 5⁰ = 1 (not 0)
- 2⁻³ = 1/8 (not −8)
- (2³)⁴ = 2¹² — multiply the powers (not add)
- 8^(2/3) = (∛8)² = 2² = 4
The 5 Index Laws
- Multiply: aᵐ × aⁿ = aᵐ⁺ⁿ (add powers)
- Divide: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract powers)
- Power of power: (aᵐ)ⁿ = aᵐⁿ (multiply powers)
- Zero power: a⁰ = 1
- Negative power: a⁻ⁿ = 1/aⁿ
Common Mistakes
- Applying laws to different bases: 2³ × 3² ≠ 6⁵ — index laws only work when the bases are the same
- Power of zero: a⁰ = 1, not 0 — any non-zero number raised to the power 0 is always 1
- Negative power means reciprocal: 2⁻³ = 1/8, not −8 — flip to get the reciprocal, then apply the positive power
- Power of a power: (aᵐ)ⁿ = aᵐⁿ — multiply the powers, do not add them
Practice questions for Index Laws
Which of these is equivalent to a³ × a⁵?
Simplify a⁵ × a³
Quick recall flashcards
Zero Power Law
a⁰ = 1
ANY number to power 0 equals 1
(except 0⁰ which is undefined)
Examples: 5⁰ = 1, 100⁰ = 1
Division Law
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
SUBTRACT the powers
Example: 5⁶ ÷ 5² = 5⁴