Question

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

Answer 1

Hint: Here we will find the square roots of each of the numbers. In the method of repeated subtraction we subtract the successive odd numbers from the given number until we get zero. The number of times we subtracted the numbers is the square root of the given number.

Complete step-by-step answer:
The given numbers are 100 and 169.
Here first we will consider the number 100.
We know that, in order to find the square root by the method of repeated subtraction we need to subtract the successive odd numbers from the given number until we get zero. The number of times we subtracted the numbers is the square root of the given number.
Hence, subtracting 1 from 100 we get:-
\[100 - 1 = 99\]
Now subtracting 3 we get:-
\[99 - 3 = 96\]
Now subtracting 5 we get:-
\[96 - 5 = 91\]
Now subtracting 7 we get:-
\[91 - 7 = 84\]
Now subtracting 9 we get:-
\[84 - 9 = 75\]
Subtracting 11 we get:-
\[75 - 11 = 64\]
Now subtracting 13 we get:-
\[64 - 13 = 51\]
Subtracting 15 we get:-
\[51 - 15 = 36\]
Now subtracting 17 we get:-
\[36 - 17 = 19\]
Now subtracting 19 we get:-
\[19 - 19 = 0\]
Now since we got zero therefore, now we can write the root of 100.
Since, we subtracted the odd numbers 10 times therefore; the square root of 100 is 10.
\[\sqrt {100} = 10\]
Now let us consider the number 169.
We know that, in order to find the square root by the method of repeated subtraction we need to subtract the successive odd numbers from the given number until we get zero. The number of times we subtracted the numbers is the square root of the given number.
Hence, subtracting 1 from 169 we get:-
\[169 - 1 = 168\]
Now subtracting 3 we get:-
\[168 - 3 = 165\]
Now subtracting 5 we get:-
\[165 - 5 = 160\]
Now subtracting 7 we get:-
\[160 - 7 = 153\]
Now subtracting 9 we get:-
\[153 - 9 = 144\]
Subtracting 11 we get:-
\[144 - 11 = 133\]
Now subtracting 13 we get:-
\[133 - 13 = 120\]
Subtracting 15 we get:-
\[120 - 15 = 105\]
Now subtracting 17 we get:-
\[105 - 17 = 88\]
Now subtracting 19 we get:-
\[88 - 19 = 69\]
Now subtracting 21 we get:-
\[69 - 21 = 48\]
Now subtracting 23 we get:-
\[48 - 23 = 25\]
Now subtracting 25 we get:-
\[25 - 25 = 0\]
Now since we got zero therefore, now we can write the root of 169.
Since, we subtracted the odd numbers 13 times therefore; the square root of 169 is 13.
\[\sqrt {169} = 13\]

Note: Students should note that the method of repeated subtraction can only be used for small numbers as it is a lengthy method. Also, it cannot be used for decimal numbers.
There are several other methods to find the square root like the prime factorization method.