Question

Find the square root of the following numbers by the repeated subtraction method.
I.16
II.49
III.64
IV.100
V.169
VI.81

Answer 1

Hint: Here in this question we have to determine the square root of the given numbers by repeated subtraction. In the repeated subtraction we are going to subtract the odd number till the given number from the given number. Suppose if we think n is a number, we have to determine the square root of n by the repeated subtraction method. So we subtract the odd numbers from 1 to n. On subtracting we get a number 0. The number of steps involved to get a result as a zero is the square root of the given number.

Complete step by step solution:
A square number can be defined as an integer that can be expressed as a product of a number with the number itself. And the number which is multiplied with itself is called the square root of the square number.
Here we have to determine the square root of a number by repeated subtraction method.
Now we solve the set of questions
I.16
The odd numbers from 1 to 16 will be 1, 3, 5,..15.
Step 1: \[16 - 1 = 15\]
Step 2: \[15 - 3 = 12\]
Step 3: \[12 - 5 = 7\]
Step 4: \[7 - 7 = 0\]
Finally we got the result as 0 at the fourth step.
Therefore the square root of 16 is 4.

II.49
The odd numbers from 1 to 16 will be 1, 3, 5,..49.
Step 1: \[49 - 1 = 48\]
Step 2: \[48 - 3 = 45\]
Step 3: \[45 - 5 = 40\]
Step 4: \[40 - 7 = 33\]
Step 5: \[33 - 9 = 24\]
Step 6: \[24 - 11 = 13\]
Step 7: \[13 - 13 = 0\]
Finally we got the result as 0 at the seventh step.
Therefore the square root of 49 is 7.

III.64
The odd numbers from 1 to 16 will be 1, 3, 5,..63.
Step 1: \[64 - 1 = 63\]
Step 2: \[63 - 3 = 60\]
Step 3: \[60 - 5 = 55\]
Step 4: \[55 - 7 = 48\]
Step 5: \[48 - 9 = 39\]
Step 6: \[39 - 11 = 28\]
Step 7: \[28 - 13 = 15\]
Step 8: \[15 - 15 = 0\]
Finally we got the result as 0 at eighth step.
Therefore the square root of 64 is 8

IV.100
The odd numbers from 1 to 16 will be 1, 3, 5,..99.
Step 1: \[100 - 1 = 99\]
Step 2: \[99 - 3 = 96\]
Step 3: \[96 - 5 = 91\]
Step 4: \[91 - 7 = 84\]
Step 5: \[84 - 9 = 75\]
Step 6: \[75 - 11 = 64\]
Step 7: \[64 - 13 = 51\]
Step 8: \[51 - 15 = 36\]
Step 9: \[36 - 17 = 19\]
Step 10: \[19 - 19 = 0\]
Finally we got the result as 0 at tenth step.
Therefore the square root of 100 is 10.

V.169
The odd numbers from 1 to 16 will be 1, 3, 5,..169.
Step 1: \[169 - 1 = 168\]
Step 2: \[168 - 3 = 165\]
Step 3: \[165 - 5 = 160\]
Step 4: \[160 - 7 = 153\]
Step 5: \[153 - 9 = 144\]
Step 6: \[144 - 11 = 133\]
Step 7: \[133 - 13 = 120\]
Step 8: \[120 - 15 = 105\]
Step 9: \[105 - 17 = 88\]
Step 10: \[88 - 19 = 69\]
Step 11: \[69 - 21 = 48\]
Step 12: \[48 - 23 = 25\]
Step 13: \[25 - 25 = 0\]
Finally we got the result as 0 at thirteenth step.
Therefore the square root of 169 is 13.

VI.81
The odd numbers from 1 to 16 will be 1, 3, 5,..81.
Step 1: \[81 - 1 = 80\]
Step 2: \[80 - 3 = 77\]
Step 3: \[77 - 5 = 72\]
Step 4: \[72 - 7 = 65\]
Step 5: \[65 - 9 = 56\]
Step 6: \[56 - 11 = 45\]
Step 7: \[45 - 13 = 32\]
Step 8: \[32 - 15 = 17\]
Step 9: \[17 - 17 = 0\]
Finally we got the result as 0 at the ninth step.
Therefore the square root of 81 is 9

Note: The square root of a number can be determined by many methods namely, prime factorisation method, repeated subtraction method, division method, number line method and average method. Here to solve this kind of problem the student must know the subtraction and the odd numbers. We need not to consider all the odd numbers. Sometimes we will get a result as 0, till that we need the odd numbers.