Question

What is the square root of $21$?

Answer 1

Hint: In the above question, we have been asked to find out the square root of $21$. Since $21$ is not a perfect square, its square root will be an irrational number. The irrational numbers are those having decimal expansions which are neither terminating nor repeating. The square roots of imperfect squares can be found out by using the long division method. Therefore, for obtaining the square root of $21$ we will use the long division method.

Complete step by step solution:
According to the question, we need to find out the square root of the number $21$. The perfect square number just before $21$ is \[16\], which is equal to the square of four. Also, the perfect square just after $21$ is $25$, which is equal to the square of five. Since the given number $21$ lies between the squares of two consecutive natural numbers, $21$ is an imperfect square.
Now, we know that the square root of an imperfect square is irrational. It can be found out using the long division method. The long division for obtaining the square root of $21$ is shown below.
\[4\overset{4.58}{\overline{\left){\begin{align}
  & \overline{21}.\overline{00} \\
 & \underline{16} \\
 & 85\overline{\left){\begin{align}
  & 500 \\
 & \underline{425} \\
 & 908\overline{\left){\begin{align}
  & 7500 \\
 & \underline{7264} \\
 & \underline{236} \\
\end{align}}\right.} \\
\end{align}}\right.} \\
\end{align}}\right.}}\]
From the above long division, we got the square root of $21$ equal to $4.58$.

Hence, the square root of $21$ is equal to $4.58$.

Note: The value of the square root of $21$, which is obtained to be equal to $4.58$, is just an approximate value. The long division shown above will never end because of which the decimal expansion will never terminate, which is the proof that the square root of $21$ is irrational.