Surds and Indices — 21 Aptitude Questions and Answers with Explanations

Surds and Indices — Questions and Answers

Surds and Indices is a rules topic: six exponent laws and two surd operations cover every question that appears. Fractional exponents are where candidates hesitate, so fix the meaning early — a^(m/n) is the nth root of a raised to the power m. Once that clicks, expressions like 16^(3/4) become a two-second calculation.

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Surds and Indices— Concepts, Formulas & Shortcuts

aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ.
a⁰ = 1 for any non-zero a; a⁻ⁿ = 1/aⁿ.
a^(m/n) = ⁿ√(aᵐ) — the denominator is the root, the numerator is the power.
√a × √b = √(ab) and √a ÷ √b = √(a/b).
Rationalise a denominator by multiplying by the conjugate: 1/√3 = √3/3.
To solve aˣ = b, express b as a power of a and equate the exponents.

Surds and Indices Practice Questions with Answers

Attempt each question first, then open the explanation. All 21 questions below are free to read and require no signup.

Q1.The value of (2³)² is:
Easy
- A12
- B32
- C64
- D128
+Show Answer & Explanation
Answer: C. 64
Explanation: (aᵐ)ⁿ = aᵐⁿ, so 2⁶ = 64.
Surds and Indices question 1 of 21
Q2.The value of 5⁰ is:
Easy
- A0
- B1
- C5
- DUndefined
+Show Answer & Explanation
Answer: B. 1
Explanation: Any non-zero number raised to the power zero equals 1.
Surds and Indices question 2 of 21
Q3.The value of √8 × √2 is:
Easy
- A2
- B4
- C8
- D16
+Show Answer & Explanation
Answer: B. 4
Explanation: √8 × √2 = √16 = 4.
Surds and Indices question 3 of 21
Q4.aᵐ × aⁿ equals:
Easy
- Aaᵐⁿ
- Baᵐ⁺ⁿ
- Caᵐ⁻ⁿ
- D(a + a)ᵐⁿ
+Show Answer & Explanation
Answer: B. aᵐ⁺ⁿ
Explanation: Multiplying powers of the same base adds the exponents.
Surds and Indices question 4 of 21
Q5.If 2ˣ = 32, then x is:
Easy
- A3
- B4
- C5
- D6
+Show Answer & Explanation
Answer: C. 5
Explanation: 32 = 2⁵, so x = 5.
Surds and Indices question 5 of 21
Q6.The value of 16^(3/4) is:
Moderate
- A4
- B8
- C12
- D64
+Show Answer & Explanation
Answer: B. 8
Explanation: Take the fourth root first: ⁴√16 = 2, then 2³ = 8.
Surds and Indices question 6 of 21
Q7.Rationalising 1/√3 gives:
Moderate
- A√3
- B√3/3
- C3√3
- D1/3
+Show Answer & Explanation
Answer: B. √3/3
Explanation: Multiply numerator and denominator by √3: √3/(√3 × √3) = √3/3.
Surds and Indices question 7 of 21
Q8.The value of 3⁴ is:
Easy
- A12
- B27
- C64
- D81
+Show Answer & Explanation
Answer: D. 81
Explanation: 3 × 3 × 3 × 3 = 81.
Surds and Indices question 8 of 21
Q9.The value of 2⁻³ is:
Easy
- A−8
- B−6
- C1/6
- D1/8
+Show Answer & Explanation
Answer: D. 1/8
Explanation: A negative exponent gives the reciprocal: 2⁻³ = 1/2³ = 1/8.
Surds and Indices question 9 of 21
Q10.The value of 27^(2/3) is:
Moderate
- A3
- B6
- C9
- D18
+Show Answer & Explanation
Answer: C. 9
Explanation: Take the cube root first: ∛27 = 3, then 3² = 9.
Surds and Indices question 10 of 21
Q11.Simplify: (x⁵ × x³)/x⁴
Moderate
- Ax²
- Bx⁴
- Cx⁶
- Dx¹²
+Show Answer & Explanation
Answer: B. x⁴
Explanation: Multiplying powers adds exponents and dividing subtracts them: x^(5 + 3 − 4) = x⁴.
Surds and Indices question 11 of 21
Q12.The value of √50 + √18 is:
Difficult
- A5√2
- B8√2
- C√68
- D68
+Show Answer & Explanation
Answer: B. 8√2
Explanation: √50 = 5√2 and √18 = 3√2, so the sum is 8√2.
Surds and Indices question 12 of 21
Q13.If 3ˣ = 81, then x equals:
Easy
- A2
- B3
- C4
- D5
+Show Answer & Explanation
Answer: C. 4
Explanation: 81 = 3⁴, so x = 4.
Surds and Indices question 13 of 21
Q14.The value of (√5 + √3)(√5 − √3) is:
Moderate
- A2
- B8
- C15
- D√15
+Show Answer & Explanation
Answer: A. 2
Explanation: Using (a + b)(a − b) = a² − b²: 5 − 3 = 2.
Surds and Indices question 14 of 21
Q15.The value of 6² is:
Easy
- A12
- B24
- C36
- D42
+Show Answer & Explanation
Answer: C. 36
Explanation: Squaring means multiplying the number by itself: 6 × 6 = 36.
Surds and Indices question 15 of 21
Q16.The value of 10⁻² is:
Easy
- A−100
- B−20
- C0.01
- D0.1
+Show Answer & Explanation
Answer: C. 0.01
Explanation: 10⁻² = 1/10² = 1/100 = 0.01.
Surds and Indices question 16 of 21
Q17.Simplify: 2⁵ ÷ 2²
Easy
- A2²
- B2³
- C2⁷
- D2¹⁰
+Show Answer & Explanation
Answer: B. 2³
Explanation: Dividing powers subtracts exponents: 2⁵⁻² = 2³.
Surds and Indices question 17 of 21
Q18.The value of 8^(1/3) is:
Moderate
- A2
- B3
- C4
- D8
+Show Answer & Explanation
Answer: A. 2
Explanation: The exponent 1/3 means the cube root, and ∛8 = 2.
Surds and Indices question 18 of 21
Q19.The value of √12 × √3 is:
Moderate
- A3
- B6
- C9
- D36
+Show Answer & Explanation
Answer: B. 6
Explanation: √12 × √3 = √36 = 6.
Surds and Indices question 19 of 21
Q20.If 5ˣ = 125, then x equals:
Easy
- A2
- B3
- C4
- D5
+Show Answer & Explanation
Answer: B. 3
Explanation: 125 = 5³, so x = 3.
Surds and Indices question 20 of 21
Q21.The value of (3²)³ is:
Moderate
- A3⁵
- B3⁶
- C3⁸
- D3⁹
+Show Answer & Explanation
Answer: B. 3⁶
Explanation: (aᵐ)ⁿ = aᵐⁿ, so 3²ˣ³ = 3⁶.
Surds and Indices question 21 of 21

Surds and Indices — Frequently Asked Questions

What does a fractional exponent mean?+

a^(m/n) is the nth root of a to the power m. So 16^(3/4) = (⁴√16)³ = 2³ = 8. Taking the root first keeps the numbers small.

Why do we rationalise the denominator?+

To remove the surd from the bottom of a fraction, making the value easier to estimate and to compare with options. Multiply numerator and denominator by the surd or by the conjugate.

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