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Surds and Indices Questions with Solutions

A handful of fixed rules for combining powers of the same base cover almost everything in this topic — the only real skill is applying them in the right order. Here’s the concept, four fully worked examples, and a few to try yourself.

The concept, quickly

  • For powers of the SAME base: aᵐ × aⁿ = aᵐ⁺ⁿ (add exponents when multiplying); aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract when dividing); (aᵐ)ⁿ = aᵐⁿ (multiply when raising a power to a power).
  • Any nonzero number to the power 0 equals 1 (a⁰ = 1). A negative exponent means "reciprocal": a⁻ⁿ = 1/aⁿ.
  • A surd is an irrational root that can’t be simplified to a whole number, like √2 or √3 (√4 = 2 is NOT a surd, since it simplifies exactly). To simplify one, pull out the largest perfect-square factor: √50 = √(25 × 2) = 5√2.
  • To "rationalize" a fraction with a surd in the denominator, multiply top and bottom by that surd (or its conjugate, if the denominator has two terms), so the denominator becomes a whole number: 1/√2 = √2/2.

Worked examples

1. Simplify: 2⁵ × 2³

Solution

  1. Same base, so add the exponents: 2⁵⁺³ = 2⁸.
  2. 2⁸ = 256.

Answer: 256

2. Simplify: (3⁴)² ÷ 3⁵

Solution

  1. (3⁴)² = 3⁸ (multiply the exponents).
  2. 3⁸ ÷ 3⁵ = 3⁸⁻⁵ = 3³ = 27.

Answer: 27

3. Simplify √72.

Solution

  1. 72 = 36 × 2, and 36 is the largest perfect-square factor of 72.
  2. √72 = √(36 × 2) = 6√2.

Answer: 6√2

4. Rationalize: 1 / (2 + √3)

Solution

  1. Multiply top and bottom by the conjugate, (2 − √3): [1 × (2 − √3)] / [(2 + √3)(2 − √3)].
  2. The denominator becomes 2² − (√3)² = 4 − 3 = 1.
  3. Result: (2 − √3) / 1 = 2 − √3.

Answer: 2 − √3

Try these yourself

Work through these the same way, then check against the answer.

1. Simplify: 5⁶ ÷ 5²

Answer: 625

2. Simplify √98.

Answer: 7√2

3. Simplify: (2³)² × 2⁻⁴

Answer: 4

Where this comes up

Surds and Indices is tested directly in the Quantitative/Numerical Aptitude section of every exam Pariksha Saathi covers — SSC CGL, SSC MTS, SSC CHSL, IBPS PO, IBPS Clerk, SBI PO and SBI Clerk.

Practice more Surds and Indices questions

This page covers the concept and a handful of worked examples. Pariksha Saathi has full topic-wise practice sets for Surds and Indices (and every other Quant/Reasoning/English topic) with instant scoring and explanations — free, no account required.

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