Indices and surds — Practice MCQs

1. Laws of indices

Trap: a^(m^n) is not (aᵐ)ⁿ. 2^(3²) = 2⁹ = 512 but (2³)² = 64.

2. Solving exponential equations

1. Write both sides with the same base (4 = 2², 8 = 2³, 27 = 3³, 125 = 5³, 0.25 = 2⁻²).
2. Equate the powers.

4ˣ⁺¹ = 8ˣ → 2²ˣ⁺² = 2³ˣ → x = 2.

3ˣ⁻¹ + 3ˣ⁺¹ = 90 → 3ˣ(1/3 + 3) = 90 → 3ˣ = 27 → x = 3.

(ASO 2021) p⁵ˣ⁺³ = 1 with p ≠ 1 → 5x + 3 = 0 → x = −3/5.

(ASO 2025) x = yᵃ, y = zᵇ, z = xᶜ → x = x^(abc) → abc = 1.

3. Surds

A surd is an irrational root like √2, ∛5. Order of a surd = the root index.

• √a × √b = √ab; √a ÷ √b = √(a/b); (√a)² = a.
• Simplify: √72 = √(36 × 2) = 6√2.
• Only like surds add: 3√2 + 5√2 = 8√2; √2 + √3 cannot combine.
• Rationalise: 1/(√a ± √b) = (√a ∓ √b)/(a − b).
• Conjugate pairs: (√a + √b)(√a − √b) = a − b.

1/(√7 − √5) = (√7 + √5)/2.

If x = 3 + 2√2, then 1/x = 3 − 2√2 and x + 1/x = 6.

Sum 1/(√1 + √2) + 1/(√2 + √3) + … + 1/(√8 + √9) = √9 − √1 = 2 (telescoping).

4. Comparing surds

Raise to a common power (LCM of the root indices). Compare √3, ∛4, ⁶√15: to the 6th power → 27, 16, 15 → √3 is largest.

Solved examples by question type

Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.

Basic level

Example 1. 2^5 ÷ 2^3 = ?
(a) 1.67   (b) 4   (c) 8   (d) 256
Answer: (b) 4
Solution: aᵐ ÷ aⁿ = aᵐ⁻ⁿ = 2^2 = 4.

Example 2. 2^7 ÷ 2^4 = ?
(a) 1.75   (b) 2,048   (c) 8   (d) 16
Answer: (c) 8
Solution: aᵐ ÷ aⁿ = aᵐ⁻ⁿ = 2^3 = 8.

Example 3. 4^12 × 4^15 ÷ 4^15 = ?
(a) 4^12   (b) 4^42   (c) 4^13   (d) 4^165
Answer: (a) 4^12
Solution: Add powers when multiplying, subtract when dividing: 12 + 15 − 15 = 12.

Example 4. 4^4 × 4^11 ÷ 4^4 = ?
(a) 4^11   (b) 4^19   (c) 4^12   (d) 4^40
Answer: (a) 4^11
Solution: Add powers when multiplying, subtract when dividing: 4 + 11 − 4 = 11.

ASO exam level

Example 5. If 81^x = 5,31,441, then x = ?
(a) 4   (b) 2   (c) 3   (d) 12
Answer: (c) 3
Solution: 81 = 3^4, 5,31,441 = 3^12. So 4x = 12 → x = 3.

Example 6. If x = y^a, y = z^b and z = x^c, then the value of abc is:
(a) 1   (b) a + b + c   (c) 0   (d) −1
Answer: (a) 1
Solution: x = y^a = z^(ab) = x^(abc). So abc = 1.

Example 7. If 3^(x + 2) = 729, then x = ?
(a) 8   (b) 5   (c) 4   (d) 6
Answer: (c) 4
Solution: 729 = 3^6, so x + 2 = 6 → x = 4.

Example 8. If 2^(x − 1) = 128, then x = ?
(a) 7   (b) 9   (c) 6   (d) 8
Answer: (d) 8
Solution: 128 = 2^7, so x − 1 = 7 → x = 8.

Example 9. (64)^(4/3) = ?
(a) 16   (b) 1,024   (c) 257   (d) 256
Answer: (d) 256
Solution: (64)^(1/3) = 4; then 4^4 = 256.

Advanced level

Example 10. Rationalise: 1/(√7 + √5) = ?
(a) (√7 − √5)/12   (b) (√7 − √5)/2   (c) (√7 + √5)/12   (d) √7 + √5
Answer: (b) (√7 − √5)/2
Solution: Multiply top and bottom by (√7 − √5): denominator = 7 − 5 = 2.

Example 11. (3/2)^(−4) = ?
(a) 16/81   (b) 81/16   (c) −81/16   (d) 8/12
Answer: (a) 16/81
Solution: A negative power flips the fraction: (2/3)^4 = 16/81.

Example 12. (2/6)^(−4) = ?
(a) 16/1296   (b) 24/8   (c) 1296/16   (d) −16/1296
Answer: (c) 1296/16
Solution: A negative power flips the fraction: (6/2)^4 = 1296/16.

Topic-wise practice for Indices and surds (Quant): Laws of indices, rationalising, comparing surds, exponent equations. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.

Question types covered

• Laws of indices; negative and fractional powers
• Solving exponent equations (find x)
• Simplifying surds
• Rationalising the denominator
• Comparing surds and finding the largest or smallest
• Unit digit and prime factors of powers