Question

Find the number of digits in the square root of each of the following numbers (without calculator).
(i) 64 (ii) 144 (iii) 4489 (iv) 27225 (v) 390625

Answer 1

 Hint: To determine the number of digits in the square root of a number, first we have to determine that the total number of digits in the given number is odd or even. If the total number of digits in the number is even then, the number of digits in square root will be determined by using the formula \[N = \dfrac{n}{2}\] and in the case of odd number of digits square root will be \[N = \dfrac{{n + 1}}{2}\] where, $n$ is the number of digits in the number.

Complete step-by-step answer:
(i) Number of digits in 64 is 2 so, \[n = 2\]
Since $n$ is even so,
Numbers of digits in square root in the square root of 64 is:
 \[
  N = \dfrac{n}{2} \\
   = \dfrac{2}{2} \\
   = 1 \\
 \]
Hence, the total number of digits in the square root of 64 is 1.

(ii) Number of digits in 144 is 3 so, \[n = 3\]
Since $n$is odd so,
Number of digits in square root of 144 is:
 \[
  N = \dfrac{{n + 1}}{2} \\
   = \dfrac{{3 + 1}}{2} \\
   = \dfrac{4}{2} \\
   = 2 \\
 \]
Hence, the total number of digits in the square root of 144 is 2.

(iii) Number of digits in 4489 is 4 so, \[n = 4\]
Since $n$ is even so,
Numbers of digits in square root of 4489 is:
\[
  N = \dfrac{n}{2} \\
   = \dfrac{4}{2} \\
   = 2 \\
 \]
Hence, the total number of digits in the square root of 4489 is 2.

(iv) Number of digits in 27225 is 5 so, \[n = 5\]
Since $n$ is odd so,
Number of digits in square root of 27225 is:
\[
  N = \dfrac{{n + 1}}{2} \\
   = \dfrac{{5 + 1}}{2} \\
   = \dfrac{6}{2} \\
   = 3 \\
 \]
Hence, the total number of digits in the square root of 27225 is 3.

(v) Number of digits in 390625 is 6 so, \[n = 6\]
Since $n$ is even so,
Number of digits in square root of 390625 is:
\[
  N = \dfrac{n}{2} \\
   = \dfrac{6}{2} \\
   = 3 \\
 \]
Hence, the total number of digits in the square root of 390625 is 3.

Note: Count the numbers of digits in the given number and determine whether even or odd. The method is applicable only for a perfect square number.