NumberFoundation + Higher★ Core topic
Powers, Roots
& Indices.
A power (or index) tells you how many times to multiply a number by itself. A root is the reverse. Seven index laws cover every GCSE question — learn them, and you'll never panic again.
I · Key definitions
Power / index
How many times you multiply a number by itself. a ⁿ.
2⁴ = 2×2×2×2 = 16
Base
The number being raised to a power.
in 5³, the base is 5
Square root
The reverse of squaring. √a asks what × itself = a?
√49 = 7
Cube root
∛a asks what × itself × itself = a?
∛125 = 5
Negative power
Means reciprocal (flip the fraction).
2⁻³ = 1/2³ = 1/8
Fractional power
Means a root. a^(1/n) = ⁿ√a.
8^(1/3) = ∛8 = 2
II · The seven index laws
aᵐ × aⁿ = aᵐ⁺ⁿ
same base × → add powers
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
same base ÷ → subtract powers
(aᵐ)ⁿ = aᵐⁿ
power to a power → multiply powers
a⁰ = 1
anything to the 0 is 1 (except 0⁰)
a⁻ⁿ = 1/aⁿ
negative power → reciprocal
a^(1/n) = ⁿ√a
unit fraction → root
a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
"root first, then power" — or the other way round
III · Powers of 2 — worth memorising
2¹
2
2²
4
2³
8
2⁴
16
2⁵
32
2⁶
64
2⁷
128
2⁸
256
2⁹
512
2¹⁰
1024
Also memorise: squares up to 15² (=225) and cubes up to 5³ (=125). These appear in almost every paper.
IV · Worked examples
Simplify using index laws
2 marks
Simplify: 2³ × 2⁵ ÷ 2⁴
i.Same base — apply the laws. 2³ × 2⁵ = 2³⁺⁵ = 2⁸
ii.2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴
iii.Evaluate: 2⁴ = 16
2³ × 2⁵ ÷ 2⁴ = 2⁴ = 16
Evaluate fractional powers
3 marks
Work out the value of 8^(2/3).
i.Split: 8^(2/3) = (8^(1/3))² — take the root first, it's easier.
ii.8^(1/3) = ∛8 = 2
iii.Square it: 2² = 4
8^(2/3) = 4
Negative & zero powers
2 marks
Evaluate: (a) 5⁰ (b) 3⁻² (c) (2/3)⁻¹
(a)Any non-zero number to the power 0 = 1
(b)3⁻² = 1/3² = 1/9
(c)Flip: (2/3)⁻¹ = 3/2
1 · 1/9 · 3/2
V · Common mistakes & examiner tips
Common mistakes
Multiplying powers when you should add. 2³ × 2⁵ = 2⁸, not 2¹⁵.
Assuming 0⁰ = 0 or 1. It's undefined at GCSE.
Negative base ambiguity. (−3)² = 9 but −3² = −9 (only the 3 is squared).
Forgetting a⁰ = 1. Not 0. Anything non-zero to the 0 is 1.
For a^(m/n): do the root, not the power, first. Smaller number, fewer mistakes.
Examiner tips
Write all index laws at the top of the page before you start.
Negative power → flip. That's the first thing to do.
Fractional power → root. Denominator is the root; numerator is the power.
Don't calculate prematurely — simplify using index laws first.
Memorise powers of 2 up to 2¹⁰ — saves time in exams.