GCSE Maths | Indices

GCSE Maths Indices Worksheet: Practice Questions by Topic

A repetition-based indices worksheet for GCSE Maths. Each section drills one index law at a time. Work through the sections in order — the aim is confident pattern recognition before tackling mixed questions elsewhere.

How do you multiply indices with the same base?

Rule: a^m × aⁿ = a^{m + n}
When the base is the same, add the powers.

Example: m⁴ × m³ = m^{4 + 3} = m⁷

(a) m² × m³
(b) m³ × m³
(c) m⁶ × m²
(d) m⁷ × m³
(e) m⁶ × m⁸
(f) m² × m
(g) m × m³
(h) m⁷ × m⁸
(i) m⁹ × m²
(j) m × m⁸
(k) m⁶ × m⁵
(l) m² × m² × m² × m²

Answers: (a) m⁵ (b) m⁶ (c) m⁸ (d) m¹⁰ (e) m¹⁴ (f) m³ (g) m⁴ (h) m¹⁵ (i) m¹¹ (j) m⁹ (k) m¹¹ (l) m⁸

How do you divide indices with the same base?

Rule: a^m ÷ aⁿ = a^{m − n}
When the base is the same, subtract the powers.

Example: n⁷ ÷ n² = n^{7 − 2} = n⁵

(a) n⁵ ÷ n²
(b) n⁸ ÷ n³
(c) n⁹ ÷ n²
(d) n⁷ ÷ n⁵
(e) n³ ÷ n
(f) n⁸ ÷ n
(g) n⁷ ÷ n⁴
(h) n⁹ ÷ n³
(i) n⁴ ÷ n⁸
(j) n ÷ n³
(k) n⁴⁵ ÷ n⁵
(l) n³ ÷ n³

Answers: (a) n³ (b) n⁵ (c) n⁷ (d) n² (e) n² (f) n⁷ (g) n³ (h) n⁶ (i) n⁻⁴ (j) n⁻² (k) n⁴⁰ (l) 1

How do you raise a power to another power?

Rule: (a^m)ⁿ = a^{m × n}
When a power is raised to another power, multiply the powers.

Example: (y⁴)³ = y^{4 × 3} = y¹²

(a) (y⁵)²
(b) (y³)²
(c) (y⁴)³
(d) (y⁵)⁴
(e) (y³)⁶
(f) (y⁷)³
(g) (y⁶)⁶
(h) (y⁹)²
(i) (y⁴)⁸
(j) (y³)⁻⁵
(k) (y⁻⁵)²

Answers: (a) y¹⁰ (b) y⁶ (c) y¹² (d) y²⁰ (e) y¹⁸ (f) y²¹ (g) y³⁶ (h) y¹⁸ (i) y³² (j) y⁻¹⁵ (k) y⁻¹⁰

How do you simplify a bracket with a coefficient raised to a power?

Rule: (ka^m)ⁿ = kⁿ × a^{m × n}
Apply the outer power to both the number and the letter.

Example: (3x²)³ = 3³ × x^{2 × 3} = 27x⁶

(a) (2x³)²
(b) (5x⁶)²
(c) (5x⁵)³
(d) (2x³)⁴
(e) (7x⁵)²
(f) (4x⁷)³
(g) (2x⁶)⁶
(h) (10x⁹)³
(i) (3x⁴)⁴

Answers: (a) 4x⁶ (b) 25x¹² (c) 125x¹⁵ (d) 16x¹² (e) 49x¹⁰ (f) 64x²¹ (g) 64x³⁶ (h) 1000x²⁷ (i) 81x¹⁶

What is any number to the power of 0?

Rule: a⁰ = 1 (for any non-zero a)
Anything to the power of 0 equals 1.

Example: 6⁰ = 1

(a) 5⁰
(b) 12⁰
(c) x⁰
(d) 100⁰
(e) (−3)⁰
(f) y⁰
(g) 7⁰
(h) (2x)⁰
(i) (1/2)⁰
(j) m⁰
(k) (4xy)⁰
(l) 1⁰

Answers: All answers = 1

What does a power of −1 mean?

Rule: a⁻¹ = 1/a
A power of −1 means the reciprocal — flip the number.

Example: 8⁻¹ = 1/8

(a) 5⁻¹
(b) 2⁻¹
(c) 10⁻¹
(d) 3⁻¹
(e) 7⁻¹
(f) 4⁻¹
(g) x⁻¹
(h) 12⁻¹
(i) 100⁻¹
(j) (1/2)⁻¹
(k) (2/3)⁻¹
(l) (3/5)⁻¹

Answers: (a) 1/5 (b) 1/2 (c) 1/10 (d) 1/3 (e) 1/7 (f) 1/4 (g) 1/x (h) 1/12 (i) 1/100 (j) 2 (k) 3/2 (l) 5/3

What does a power of 1/2 mean?

Rule: a^1/2 = √a
A power of 1/2 means the square root.

Example: 25^1/2 = √25 = 5

(a) 9^1/2
(b) 16^1/2
(c) 25^1/2
(d) 36^1/2
(e) 49^1/2
(f) 64^1/2
(g) 81^1/2
(h) 100^1/2
(i) 121^1/2
(j) 144^1/2
(k) 4^1/2
(l) 400^1/2

Answers: (a) 3 (b) 4 (c) 5 (d) 6 (e) 7 (f) 8 (g) 9 (h) 10 (i) 11 (j) 12 (k) 2 (l) 20

What does a power of 1/3 mean?

Rule: a^1/3 = ³√a
A power of 1/3 means the cube root.

Example: 64^1/3 = ³√64 = 4

(a) 8^1/3
(b) 27^1/3
(c) 64^1/3
(d) 125^1/3
(e) 1000^1/3
(f) 1^1/3
(g) 216^1/3
(h) 343^1/3
(i) 512^1/3
(j) 729^1/3
(k) (1/8)^1/3
(l) (1/27)^1/3

Answers: (a) 2 (b) 3 (c) 4 (d) 5 (e) 10 (f) 1 (g) 6 (h) 7 (i) 8 (j) 9 (k) 1/2 (l) 1/3

How do you evaluate a power of 3/2 or 2/3?

Rule: a^m/n = (ⁿ√a)^m
Do the root first (from the denominator), then the power (from the numerator). Rooting first keeps the numbers small.

Example: 8^2/3 = (³√8)² = 2² = 4

Part 1 — powers of 3/2:

(a) 4^3/2
(b) 9^3/2
(c) 16^3/2
(d) 25^3/2
(e) 36^3/2
(f) 100^3/2

Part 2 — powers of 2/3:

(g) 8^2/3
(h) 27^2/3
(i) 64^2/3
(j) 125^2/3
(k) 1000^2/3
(l) 216^2/3

Answers: (a) 8 (b) 27 (c) 64 (d) 125 (e) 216 (f) 1000 (g) 4 (h) 9 (i) 16 (j) 25 (k) 100 (l) 36

How do you evaluate negative fractional indices?

Rule: a^−m/n = 1 / a^m/n
The minus sign makes it a reciprocal. Then evaluate the fractional power as before: root first, then power.

Example: 9^−1/2 = 1 / 9^1/2 = 1/3

(a) 4^−1/2
(b) 9^−1/2
(c) 16^−1/2
(d) 25^−1/2
(e) 8^−1/3
(f) 27^−1/3
(g) 64^−1/3
(h) 125^−1/3
(i) 4^−3/2
(j) 9^−3/2
(k) 8^−2/3
(l) 27^−2/3

Answers: (a) 1/2 (b) 1/3 (c) 1/4 (d) 1/5 (e) 1/2 (f) 1/3 (g) 1/4 (h) 1/5 (i) 1/8 (j) 1/27 (k) 1/4 (l) 1/9