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Square Root & Cube Root Interview Questions

37 Questions

Square Root & Cube Root

The square root of a number is the value that, when multiplied by itself, gives the original number. The cube root of a number is the value that, when multiplied by itself three times, gives the original number. Both are denoted by the radical sign (√ for square root, ³√ for cube root) and are essential tools for simplifying surds, solving equations, and speeding up calculations in quantitative aptitude.

  • The square of a number is obtained by multiplying the number by itself. For example, the square of 6 is 6 × 6 = 36.
  • The square root of a number is that number which, when multiplied by itself, equals the given number. For example, √81 = 9 because 9 × 9 = 81.
  • The square root symbol is √ (or ˣ√ for the xth root). By convention, √1 = 1.
  • Every positive number has two square roots (one positive, one negative), but in aptitude problems we generally take the positive root unless stated otherwise.

Example

√64 = 8, because 8 × 8 = 64.

  • Step 1: Express the number as a product of its prime factors.
  • Step 2: Group the identical factors into pairs.
  • Step 3: Take one factor from each pair and multiply them together — this product is the square root.
  • This method works cleanly only when the number is a perfect square, i.e., every prime factor appears an even number of times.

Example

4761 = 23 × 23 × 3 × 3 = 23² × 3². So √4761 = 23 × 3 = 69.

  • Used for large numbers whose prime factors are hard to find.
  • Step 1: Starting from the right, split the digits into pairs (periods) of two.
  • Step 2: Find the largest number whose square is less than or equal to the first period; this is the first digit of the quotient.
  • Step 3: Subtract, bring down the next period, then double the quotient so far to form the start of the next divisor.
  • Step 4: Find a digit to complete the divisor and quotient such that (divisor × digit) is less than or equal to the current remainder-with-period.
  • Step 5: Repeat until all periods are used — the final quotient is the square root.

Example

For 226576, splitting gives periods 22|65|76. Working through the steps gives a quotient of 476, so √226576 = 476.

  • For a decimal, mark off periods of two digits both to the left and right of the decimal point before applying the long division method.
  • If a decimal has an odd number of decimal places, add a zero to make it even before finding the square root — a decimal with an odd number of decimal places has no exact square root.
  • For a fraction with a perfect-square denominator, find the square root of the numerator and denominator separately: √(a/b) = √a/√b.
  • If the denominator isn't a perfect square, either convert the fraction to a decimal first, or multiply numerator and denominator by a suitable number to make the denominator a perfect square.

Example

√(2704/49) = √2704/√49 = 52/7.

  • The cube of a number is obtained by multiplying the number by itself three times. For example, 3³ = 3 × 3 × 3 = 27.
  • The cube root of a number x, written ³√x, is the number whose cube equals x.
  • Cube roots can be negative: the cube root of a negative number is negative, since (−a)³ = −a³.

Example

³√64 = 4, because 4 × 4 × 4 = 64.

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