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Find the square root of 100 and 169 by the method of repeated subtraction.

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Hint: Here, we will use the method of repeated subtraction i.e. .., subtracting successive odd numbers from the square.

As, we know that the method of repeated subtraction means subtracting the successive odd
 numbers from the square number, till you get zero.
Now, to find the square root of 100, we will first subtract 1 from 100 and with the obtained
 difference we will subtract it from the next odd number i.e.., 3 and this process will continue
 till we get the zero. The step number at which we get ‘0’ as difference will be the square root of 100.So, it will be as follows:
$
  Step1:100 - 1 = 99 \\
  Step2:99 - 3 = 96 \\
  Step3:96 - 5 = 91 \\
  Step4:91 - 7 = 84 \\
  Step5:84 - 9 = 75 \\
  Step6:75 - 11 = 64 \\
  Step7:64 - 13 = 51 \\
  Step8:51 - 15 = 36 \\
  Step9:36 - 17 = 19 \\
  Step10:19 - 19 = 0 \\
 $
As, we got ‘0’ in step10, the square root of 100 is 10 i.e..,$\sqrt {100} = 10$.
Similarly let us find the square root of 169, we get
$
  Step1:169 - 1 = 168 \\
  Step2:168 - 3 = 165 \\
  Step3:165 - 5 = 160 \\
  Step4:160 - 7 = 153 \\
  Step5:153 - 9 = 144 \\
  Step6:144 - 11 = 133 \\
  Step7:133 - 13 = 120 \\
  Step8:120 - 15 = 105 \\
  Step9:105 - 17 = 88 \\
  Step10:88 - 19 = 69 \\
  Step11:69 - 21 = 48 \\
  Step12:48 - 23 = 25 \\
  Step13:25 - 25 = 0 \\
$
As, we got ‘0’ in step 13, the square root of 169 is $13$i.e..,$\sqrt {169} = 13$.
Hence, the square root of 100 is 10 and 169 is 13.
Note: As, the sum of first $n$ odd natural numbers is ${n^2}$.We will use this property for
 finding the square root by the repeated subtraction. The Step number at which the
 difference becomes ‘0’ will be the square root of the given number.
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