Aptitude Preparation

Square Root and Cube Root

Aptitude Preparation Square Root and Cube Root

Square Root and Cube Root: In mathematics, the square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16. Every non-negative real number x has a unique non-negative square root, called the principal square root, which is denoted by √x, where the symbol √  is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by √9 = 3, because 32 = 3 ⋅ 3 = 9 and 3 is non-negative.

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  1.   When we multiply a number by itself three times, the product so obtained is called the perfect cube of that number.
  2. There are only 10 perfect cubes from 1 to 1000.
  3. Cubes of even numbers are even and those of odd numbers are odd
  4. The cube of a negative number is always negative.
  5. If the digit in one�fs place of a number is 0, 1, 4, 5, 6 or 9, then its cube will end in the same digit.
  6. If the digit in one�fs place of a number is 2, then its cube will end in 8 and vice-versa.
  7. If the digit in one�fs place of a number is 3, then its cube will end in 7 and vice-versa.
  8. If the prime factor of a number can not be made into groups of 3, it is not a perfect square.
  9. The symbol 3�� denotes the cube root of a number.

 

  •  Square of a number.
    If a natural number m can be expressed as n2 (where n is a natural number), then m is the square root or perfect square.
                 i.e. if m = n2          (m, n – natural numbers)
                 E.g. 81 = 3 × 3 × 3 × 3
                     = 32 × 32 = (3 × 3)2 = 92
                 Hence, 9 is the square root of 81.
     
  •  Properties of Square Root.
    Below is the table that has squares of numbers from 1 to 10.
    If we see the above results carefully, we can conclude that numbers ending with 0, 1, 4, 5, 6, or 9 at units place are perfect squares None of these end with 2, 3, 7 or 8, So numbers that end with 2, 3, 7, 8 are not perfect squares.
              Thus, numbers like 122, 457, 183, 928 are not perfect squares.
     
  • One’s digit in square of a number.
    (1). The ones digit in the square of number can be determined if the ones digit of the number is known.
    (2). The number of zeros at the end of a perfect square is always even and double the number of zeros at the end of the number
              E.g.
              
              Double zero {7002 = 490000} four zero (even)
    (3). The square of an even number is always an even number and square of an odd number is always an odd number.
              E.g.
                   
     
  • Interesting patterns of Square Root.

    Number between square numbers
    There are ‘2a’ non perfect square numbers between the square of two Consecutive natural numbers n + (n + 1)
                   Between 22 = 4 & 32 = 9 → 5, 6, 7, 8
                        2 × 2 = 4 non square numbers
                   Between
                        32 = 9 & 42 = 16 → 10, 11, 12, 13, 14, 15
                        2 × 3 = 6 non square numbers.
    Adding Consecutive odd numbers.
    So, we can conclude that the sum of first in odd natural numbers in n2 or, we can say if the number is a square number, it has to be the sum of successive odd numbers.
            E.g. 36
            Successively substract 1, 3, 5, 7 … from 36
            36 – 1 = 35
            35 – 3 = 32
            32 – 5 = 27
            27 – 7 = 20
            20 – 9 = 11
            11 – 11 = 0
            ∴ 36 is a perfect square