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ABHYAS PREP: ADVANCED QUANT · SQUARE & CUBE ROOTS

Square and cube roots by the unit-digit method

The last digit of a perfect square or cube narrows down the last digit of its root; bracketing the leading digits between two known squares or cubes fixes the rest.

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Worked examples

Example 1 · level 1

Meera arranges 11,236 chairs in a perfect square for a school function. How many chairs are in each row?

  1. 96
  2. 104
  3. 106
  4. 116
Show solution
  1. Unit digit 6 → root ends in 4 or 6.
  2. 112 lies between 10² and 11², so the root is 100-something.
  3. Choose between 104 and 106: 105² = 11,025 < 11,236, so the root is 106.
  4. Check: 106² = 11,236.

Answer: 106

Example 2 · level 2

Aditya stacks 2,62,144 identical small cubes into one big solid cube. How many small cubes lie along each edge?

  1. 54
  2. 64
  3. 66
  4. 74
Show solution
  1. Last digit 4 → root ends in 4.
  2. 262 lies between 6³ and 7³ → tens digit 6.
  3. Root = 64. Check: 64³ = 2,62,144.

Answer: 64

Example 3 · level 3

Rohan needs √92.16 for a physics answer. What is it?

  1. 0.96
  2. 9.4
  3. 9.6
  4. 96
Show solution
  1. √9,216 = 96 (unit digit and bracketing).
  2. 92.16 = 9,216/100, so √92.16 = 96/10 = 9.6.

Answer: 9.6

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