Powers, Roots & Index Laws
SPEC N6, N7 • NUMBER
The seven index laws. Square and cube numbers to memorise. Why 2⁻³ is not −8. And the Higher-tier fractional-index trick that keeps your numbers small.
Key Definitions
Five terms underpin every index question in the spec.
Power / Index
The number of times a base is multiplied by itself. In 3⁴, the base is 3 and the index is 4. So 3⁴ = 3 × 3 × 3 × 3 = 81.
Square Root
A number that multiplies by itself to give the original. √49 = 7. Unless stated otherwise, take the positive root.
Cube Root
∛x is the number whose cube is x. ∛27 = 3. Unlike square roots, cube roots of negative numbers exist and are negative: ∛(−64) = −4.
Negative Index
a⁻ⁿ = 1/aⁿ. A negative index means reciprocal — never a negative number. 2⁻³ = 1/8.
Fractional Index Higher
a^(1/n) = ⁿ√a. For a mixed fraction: a^(m/n) = (ⁿ√a)^m — root first, then power.
Memorise: Squares and Cubes
These come up constantly on both papers — and on Paper 1 you can't look them up.
Squares — 1² to 15²
1² = 12² = 43² = 94² = 165² = 256² = 367² = 498² = 649² = 8110² = 10011² = 12112² = 14413² = 16914² = 19615² = 225
Cubes — 1³ to 10³
1³ = 12³ = 83³ = 274³ = 645³ = 1256³ = 2167³ = 3438³ = 5129³ = 72910³ = 1000
Why it matters
Recognising that 196 is 14² (not just "around 200") lets you square-root instantly. Same for cube roots — spotting 125 as 5³ is the fastest way to simplify.
The Seven Index Laws
Every index question reduces to one or more of these seven rules.
aᵐ × aⁿ = aᵐ⁺ⁿ
Multiplication — add indices
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Division — subtract indices (top minus bottom)
(aᵐ)ⁿ = aᵐⁿ
Power of a power — multiply indices
a⁰ = 1
Zero index — anything (non-zero) to the 0 is 1
a⁻ⁿ = 1/aⁿ
Negative index — reciprocal
a^1/n = ⁿ√a
Fractional index = rootHigher
(ab)ⁿ = aⁿ · bⁿ
Power of a product — each factor gets the power
Law 1 — Multiplication: Add Indices
aᵐ × aⁿ = aᵐ⁺ⁿ
The rule in one line
Same base, add the indices. The base stays the same.
Examples
5² × 5³ = 5²⁺³ = 5⁵4³ × 4⁷ = 4³⁺⁷ = 4¹⁰x⁴ × x = x⁴⁺¹ = x⁵
The classic trap
Students write 5² × 5³ = 25⁵. Wrong. The base stays 5 — you don't multiply the bases. 5² × 5³ = 5⁵ = 3125, not 25⁵ which is much bigger.
Laws 2 & 3 — Division and Power of a Power
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Law 2 — Division: Subtract Indices
Top minus bottom — always. x⁸ ÷ x³ = x⁵, not x⁻⁵.
12⁸ ÷ 12³ = 12⁵ | 2⁹ ÷ 2⁵ = 2⁴
(aᵐ)ⁿ = aᵐⁿ
Law 3 — Power of a Power: Multiply Indices
Not add — multiply.
(x²)³ = x^2×3 = x⁶ (not x⁵) | (7³)⁵ = 7¹⁵
Remember the trigger words
"Times" and "multiply" → add indices. "Power of a power" (or bracket around a power) → multiply indices.
Laws 4 & 7 — Zero Index and Power of a Product
a⁰ = 1 (for any a ≠ 0)
Law 4 — Zero Index
Anything (except 0) to the power 0 is 1.
6⁰ = 1 | 1⁰ = 1 | 7223⁰ = 1 | (−5)⁰ = 1
(ab)ⁿ = aⁿ · bⁿ
Law 7 — Power of a Product: Every Factor Gets the Power
When a bracket contains a product, each factor (including the coefficient) is raised to the power.
(2x)³ = 2³ · x³ = 8x³ (not 2x³)
(4a³b)² = 4² · (a³)² · b² = 16a⁶b² (the 4 becomes 16)
The classic coefficient trap
Students write (4a³b)² = 4a⁶b², forgetting the 4². Every factor inside the bracket gets squared — the coefficient, every variable, every index.
Law 5 — Negative Index = Reciprocal
a⁻ⁿ = 1/aⁿ
The rule
A negative index flips the base onto the bottom of a fraction. It does not make the answer negative.
Worked examples
2⁻³ = 1/2³ = 1/85⁻² = 1/5² = 1/25
a⁻¹ = 1/a (the "reciprocal" of a)
(5/9)⁻¹ = 9/5 (flips the fraction)
Never write 2⁻³ = −8
This is the single most common mistake with indices. The minus sign on the index flips, it doesn't negate.
Law 6 — Fractional Index = Root Higher
a^1/n = ⁿ√a | a^m/n = (ⁿ√a)ᵐ
The rule — root first, then power
When the index is a fraction m/n:
Take the n-th root first — the denominator of the fraction tells you which root.
Then raise to the m-th power — the numerator tells you which power.
Worked example: 8^2/3
Step 1: Root first — ∛8 = 2
Step 2: Power second — 2² = 4
8^2/3 = 4
Why root first
If you power first, numbers explode. 16^3/4 root-first: ⁴√16 = 2, then 2³ = 8. Power-first: 16³ = 4096, then ⁴√4096 — nightmare without a calculator.
01MCE Worked Example 3 MARKS
Simplify fully:
(2x²y)³ × 3xy² ÷ 4x⁵y²
•
MMethod. Apply the index laws in order: power of a product (Law 7), then multiplication (Law 1), then division (Law 2).
•
CCalculate.
Step 1 — Power of a product:(2x²y)³ = 2³ · (x²)³ · y³ = 8x⁶y³
Step 2 — Multiply by 3xy²: coefficients 8 × 3 = 24; add x indices 6+1 = 7; add y indices 3+2 = 5.
→ 24x⁷y⁵
Step 3 — Divide by 4x⁵y²: coefficients 24 ÷ 4 = 6; subtract x indices 7−5 = 2; subtract y indices 5−2 = 3.
→ 6x²y³
•
EEvaluate.Answer: 6x²y³
Mark scheme: M1 for correctly expanding the bracket (including squaring the 2); M1 for correct multiplication with indices added; A1 for the final simplified answer.
Weak vs Strong: Negative Index
Evaluate 2⁻³
Weak student
"Negative index, so the answer is negative."
2⁻³ = −8 ✘
Interpreted the minus sign as affecting the result, not the index.
Strong student
"Negative index means reciprocal. Flip onto a fraction."
2⁻³ = 1/2³ = 1/8 ✔
Applied Law 5 correctly. A negative index always gives a positive result (for positive bases).
The mantra
"The minus flips. It doesn't negate."
Examiner Alert: Negative Index ≠ Negative Number
What examiners see
Students write 2⁻³ = −8, 5⁻² = −25, or x⁻¹ = −x. All wrong. The minus on the index means reciprocal, never a change of sign.
What to write instead
Always rewrite as a fraction first:
2⁻³ = 1/2³ = 1/8
5⁻² = 1/5² = 1/25
x⁻¹ = 1/x
When the answer really is negative
Only if the base itself is negative. (−2)³ = −8 (negative base, odd power). But 2⁻³ has a positive base — so the reciprocal is positive.
Examiner Alert: Fractional Index Mis-order Higher
What examiners see
On 16^3/4, students compute 16³ = 4096 first, then try to take the fourth root. Without a calculator, this is near-impossible — and often abandoned.
What to write instead
Root first, then power.
16^3/4 = (⁴√16)³ = 2³ = 8
The denominator of the fraction tells you the root. The numerator tells you the power. Root-first keeps every intermediate number small.
Mark scheme insight — Nov23 WRE
AQA examiner reports explicitly flag this: students who squared or cubed first "rarely made further progress" — the number became too large to handle.
Common Errors
Students often writeCorrectWhy it matters
2⁰ = 02⁰ = 1Zero index rule (Law 4)
5² × 5³ = 25⁵= 5⁵Base stays the same, add indices
(x²)³ = x⁵= x⁶Power of a power — multiply, not add
9^1/2 = 4.5= 3 (√9)Fractional index = root, not halve
(2x)³ = 2x³= 8x³Coefficient must be cubed too
2⁻³ = −8= 1/8Negative index = reciprocal
Practice Question 1
Simplify 7³ × 7⁵. Circle your answer.1 MARK
7¹⁵ 7⁸ 49¹⁵ 49⁸
(Classic AQA Paper 1 style — "circle your answer" with 4 options.)
1
Same base (7), multiplication → add indices (Law 1).
2
7³ × 7⁵ = 7³⁺⁵ = 7⁸
Answer: 7⁸
Distractors explained:
•
7¹⁵ — multiplied the indices (would be law 3)
•
49¹⁵ — both errors: multiplied base and multiplied indices
•
49⁸ — multiplied base (classic " 5² × 5³ = 25⁵ " trap)
Practice Question 2
Simplify, leaving your answer in index form:2 MARKS
(3⁸ × 3) ÷ 3⁴
1
Note: 3 = 3¹. Combine the numerator using Law 1.
2
3⁸ × 3¹ = 3⁹
3
Now divide using Law 2: 3⁹ ÷ 3⁴ = 3⁹⁻⁴ = 3⁵
Answer: 3⁵
Mark scheme: M1 for combining the numerator into 3⁹ (or for subtracting powers correctly); A1 for the final answer in index form. If the student writes "243" (which equals 3⁵), the mark-scheme instruction "in index form" means they lose the accuracy mark.
Practice Question 3 Higher
Given that
81^−3/4 = 3ⁿ
find the value of n. You must show your working.3 MARKS
1
Rewrite 81 as a power of 3. 81 = 3⁴.
2
Substitute: (3⁴)^−3/4 = 3ⁿ
3
Apply Law 3 (power of a power — multiply indices):
3^4 × (−3/4) = 3⁻³
4
Compare both sides — same base, so indices must be equal.
Answer: n = −3
Mark scheme: M1 for writing 81 as 3⁴; M1 for applying the power-of-a-power correctly; A1 for n = −3. Without the base conversion step, no method marks are awarded.
Summary
The Seven Index Laws
1
aᵐ × aⁿ = aᵐ⁺ⁿ — add
2
aᵐ ÷ aⁿ = aᵐ⁻ⁿ — subtract (top minus bottom)
3
(aᵐ)ⁿ = aᵐⁿ — multiply
4
a⁰ = 1
5
a⁻ⁿ = 1/aⁿ — reciprocal (not negative)
6
a^1/n = ⁿ√a [Higher]
7
(ab)ⁿ = aⁿ · bⁿ — every factor gets the power
Must-Memorise Numbers
Squares to 15²: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Cubes to 10³: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
The Two Big Traps
2⁻³ ≠ −8 . It's 1/8 . Negative index = reciprocal.
For fractional indices, root first, then power . Keeps numbers small. Critical on Paper 1 at Higher.
Ready for 1.05
Next: Fractions — Four Operations — common denominators, keep-change-flip, mixed ↔ improper conversions for Paper 1 without a calculator.
Next: Key Definitions