Question

Find the square root of the following: 156.25.

Answer 1

Hint: Hint Use the long division method to calculate the square root. Or you may keep the decimal point aside and prime factorize 15625 to get the square root and then apply the decimal point to obtain the final answer.

Complete step by step answer:
We are given the decimal number 156.25.
We need to compute the square root of this decimal number.
The standard way of solving this problem is using the long division method.
The below steps need to be followed to calculate the square root of a number using the long division method.
1) Group the digits of the dividend in pairs from the right to the left and mark each pair using a bar over it.
If there are odd no. of digits in the number, then the left most number will not form a pair and should be considered alone.
2) Consider the greatest number whose square is less than or equal to the number formed by the first group.
3) Use this number as a divisor and as a quotient of the first group and find the remainder.
4) Consider the second group from the left along with the remainder from the previous step to obtain the new dividend.
5) Twice the first divisor is to be taken as the new divisor.
6) Compute the second divisor using step 2) and repeat the remaining steps to continue the division.
7) Add a decimal point in the quotient as soon as the integral part of the dividend is exhausted.
8) Repeat the process until the remainder becomes zero. If the digits in the dividend are exhausted and the remainder is still not zero, then to continue the division add a pair of zeros and repeat the above steps.
Now, let’s apply these steps on the given decimal number 156.25.

Thus, we have the square root of 156.25 is 12.5.

Note: Alternate method to compute the square root of 156.25:
Consider the number 156.25 without its decimal point. i.e. 15625
Prime factorization of 15625:
\[
  15625 = 5 \times 3105 \\
   = 5 \times 5 \times 625 \\
   = 5 \times 5 \times 5 \times 125 \\
   = 5 \times 5 \times 5 \times 5 \times 25 \\
   = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \\
 \]
There are 3 pairs of 5 in the prime factorization.
Therefore, the square root of $15625 = 5 \times 5 \times 5 = 125$
Applying the decimal point, we get the square root of 15625 as 12.5