Square Roots & Cube Roots

Square Roots & Cube Roots:

Square Roots:-If a²=b,we say that the square root of b is a.i.e. √b = a.
                       3× 3 = 9,i.e.√9 = 3.
Cube Roots:- To find the cube root of a number, we want to find a number that
                       when multiplied by itself three times gives we the original number.
                        cube root of a number x is denoted by ∛x
                         In other words, to find the cube root of number, we want to find
                         the number that when we used multiplication three times. 
                         The cube root of 8 is 2, 
                         because  2 × 2 × 2 = 8. 
Methods for finding Square Roots:-
(1) Prime Factorization Method:

                  To find the square root of a number by using the prime factorization method

       when a given number is a perfect square:

Step I: Resolve the given number into prime factors. 

Step II: Make pairs of similar factors. 

Step III: Take the product of prime factors, choosing one factor out of every pair.

 Example :-The square root of 324 by prime factorization.

                       324 = 2 × 2 × 3 × 3 × 3 × 3

                    √324 = √(2 × 2 × 3 × 3 × 3 × 3)

                            = 2 × 3 × 3

                   Hence √324 = 18

     Example :-The square root of 6084 by prime factorization.

                     6084 = 2 × 2 × 3 × 3 × 13 × 13

                    √324 = √(2 × 2 × 3 × 3 × 13 × 13)

                            = 2 × 3 × 13

                 Hence √6084 = 78

(2) Long Division Method:

Step I: Group the digits in pairs of two, starting with the digit in the units place. 

            Each pair and the remaining digit is not in pairs (if any) is called a period. 

Step II: Think of the largest number whose square is equal to or just less than the 
              first period.Take this number as the divisor and also as the quotient. 

Step III: Subtract the product of the divisor and the quotient from the first period 

             and bring down the next period to the right of the remainder. 

            This becomes the new dividend. 

Step IV: Now, the new divisor is obtained by taking two times the quotient and 

              annexing with it a suitable digit which is also taken as the next digit of the
              quotient,  chosen in such a way that the product of the new divisor and this
              digit is equal to  or just less than the new dividend. 

Step V: Repeat steps (II), (III) and (IV) till all the periods have been taken up.

             Now, the quotient so obtained is the required square root of the given number.

Example :-The square root of 324 by long division method .      

                             

Example :-The square root of 16384 by long division method .   

Methods for finding Cube Roots:-
(1) Prime Factorization Method:

                  To find the cube root of a number by using the prime factorization method

       when a given number is a perfect cube:

Step I: Resolve the given number into prime factors. 

Step II: Make pairs of three of same prime number.             

        Step III: Choosing one out of three of the same prime factors.

Example :-Find the cube root of a 2744.

                    2744 = 2 × 2 × 2 × 7 × 7 × 7

                    ∛(2744) = ∛(2 × 2 ×2 × 7 × 7 ×7)

                            = 2 × 7

                   Hence ∛2744 = 14.

Methods for finding Square Roots short trick :-

N	N2
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144

If we have to find square root of perfect square

 number of 4 to 5 digits

Group the digits in pairs of two,

 starting with the digit in the units place. 

 Each pair and the remaining digit is not in

 pairs (if any) is called a period. 

                      Ans.    First digit            Second digit

     (1)   √(156 25)                         12                                     5

      (2)   √(60 84)                           7                                  2/8

     Since last two digits of 6084 is 84 in unit place will be 

     either 2 or 8

      7²=49 is less than 60 hence in tens place digit is 7.

     Now product of 7 and 7+1=7*8=56 

     which is less than 60 hence in unit place digit will be 8.

Example :-Find the square rot of 53824.

             Ans.                       First digit            Second digit

     (1)   √(538 24)                        23                                    2/8

     Since last two digits of 53824 is 24 in unit place will be 

     either 2 (2²=4) or 8 (8²=64).

     23²=529<538 

    Now product of 23 and 23+1=23*24=552 

     which is greater than 538 hence in unit place digit will be 2.

     Hence √(538 24) = 232

 Methods for finding Cube Roots short trick :-

N	N3
1	1
2	8
3	27
4	64
5	125
6	216
7	343
8	512
9	729
10	1000

If we have to find cube root of perfect cube number of 6 digits

Group the digits in pairs of three, starting with the digit in the units place. 

 Each pair and the remaining digit is not in pairs (if any) is called a period. 

                        Ans.    First digit                       Second digit

     (1)   ∛ (12 167)                         2                                           3

     (2)   ∛ (29 791)                         3                                          1

Since last three digits of 12167 is 167 in unit place will be 7

      2³=8 is less than 12 hence in tens place digit is 2.

Since last three digits of 29791 is 791 in unit place will be 1

      3³=27 is less than 29 hence in tens place digit is 3.