Quantitative Aptitude

Surds and Indices: Practice Questions with Answers

15 practice questions on Surds and Indices, with answers; most also have an explanation or a worked solution. Try each one before you open its answer.

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1. Q.1

   Simplify: 2⁵ × 2³

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   Answer: 256

   Solution:

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

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2. Q.2

   Simplify √72.

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   Answer: 6√2

   Solution:

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

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3. Q.3

   Which is greater: 2√5 or 3√3?

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   Answer: 3√3 is greater

   Solution:

   1. Square both to remove the roots: (2√5)² = 4 × 5 = 20. (3√3)² = 9 × 3 = 27.
   2. Since 27 > 20, 3√3 is greater.

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4. Q.4

   Simplify: 8^(2/3)

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   Answer: 4

   Solution:

   1. 8^(2/3) = (cube root of 8)², since the denominator of the fractional exponent is the root.
   2. Cube root of 8 = 2, so the result is 2² = 4.

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5. Q.5

   Simplify: 3√5 + 2√5 − √5

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   Answer: 4√5

   Solution:

   1. All three terms are like surds (same number, 5, under the root), so combine their coefficients.
   2. (3 + 2 − 1)√5 = 4√5.

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6. Q.6

   Rationalize: 1 / (2 + √3)

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   Answer: 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.

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7. Q.7

   Simplify: (3⁴)² ÷ 3⁵

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   Answer: 27

   Solution:

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

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8. Q.8

   Solve for x: 2^(x+1) = 32

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   Answer: x = 4

   Solution:

   1. Write 32 as a power of 2: 32 = 2⁵.
   2. Now both sides have the same base, so equate the exponents: x + 1 = 5.
   3. x = 4.

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9. Q.9

   Simplify √98.

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   Answer: 7√2

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10. Q.10

    Simplify: 4√3 + 5√3 − 2√3

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    Answer: 7√3. Like surds: (4+5−2)√3.

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11. Q.11

    Which is greater: 4√2 or 3√3?

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    Answer: 4√2 is greater. (4√2)² = 32 and (3√3)² = 27; 32 > 27.

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12. Q.12

    Simplify: 27^(2/3)

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    Answer: 9. Cube root of 27 is 3, then 3² = 9.

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13. Q.13

    Solve for x: 3^(2x−1) = 243

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    Answer: x = 3. 243 = 3⁵, so 2x−1 = 5.

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14. Q.14

    Simplify: 5⁶ ÷ 5²

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    Answer: 625

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15. Q.15

    Simplify: (2³)² × 2⁻⁴

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    Answer: 4

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