Question

By mere observation find the number of digits in the square roots of the following numbers.
A) 14641
B) 390625
C) 3136
D) 30625

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 the square root will be determined by using the formula $N = \dfrac{n}{2}$ and in the case of an odd number of digits, the square root will be $N = \dfrac{{n + 1}}{2}$ where $n$ is the number of digits in the number.

Complete step-by-step solution:
A) Given: - 14641
The number of digits in 14641 is 5 so,
$ \Rightarrow n = 5$
Since $n$ is odd. Then,
$ \Rightarrow N = \dfrac{{n + 1}}{2}$
Substitute the value,
$ \Rightarrow N = \dfrac{{5 + 1}}{2}$
Add the term in the numerator,
$ \Rightarrow N = \dfrac{6}{2}$
Cancel out the common factors,
$\therefore N = 3$
Hence, the total number of digits in the square root of 14641 is 3.
B) Given: - 390625
The number of digits in 390625 is 6 so,
$ \Rightarrow n = 6$
Since $n$ is even. Then,
$ \Rightarrow N = \dfrac{n}{2}$
Substitute the value,
$ \Rightarrow N = \dfrac{6}{2}$
Cancel out the common factors,
$\therefore N = 3$
Hence, the total number of digits in the square root of 390625 is 3.
C) Given: - 3136
The number of digits in 3136 is 4 so,
$ \Rightarrow n = 4$
Since $n$ is even. Then,
$ \Rightarrow N = \dfrac{n}{2}$
Substitute the value,
$ \Rightarrow N = \dfrac{4}{2}$
Cancel out the common factors,
$\therefore N = 2$
Hence, the total number of digits in the square root of 3136 is 2.
D) Given: - 30625
The number of digits in 30625 is 5 so,
$ \Rightarrow n = 5$
Since $n$ is odd. Then,
$ \Rightarrow N = \dfrac{{n + 1}}{2}$
Substitute the value,
$ \Rightarrow N = \dfrac{{5 + 1}}{2}$
Add the term in the numerator,
$ \Rightarrow N = \dfrac{6}{2}$
Cancel out the common factors,
$\therefore N = 3$
Hence, the total number of digits in the square root of 30625 is 3.

Note: Count the numbers of digits in the given number and determine whether even or odd. Then use the formula according to it. The students must know that this method is applicable only for a perfect square number.