Question

What is the square root of $14$?

Answer 1

Hint: The given number in the above question, which is equal to $14$, is an imperfect square since it lies between the squares of two natural numbers, three and four. So the square root of the given number will lie between three and four. For computing the approximate decimal expansion of the square root of $14$, we have to use the long division method.

Complete step by step solution:
The number given in the above question is $14$, whose square root has to be determined. The perfect square numbers just before the given number is equal to nine, which is equal to the square of three. And the perfect square number just after the given number is equal to sixteen, which is equal to the square of four. Since $14$ lies between $9$ and $16$, which are equal to the squares of two consecutive natural numbers, three and four, it must be an imperfect square. And we know that the square root of an imperfect square is irrational, which can be determined by using the long division method. The long division method for the calculation of the square root of $14$ is as below.
\[3\overset{3.74}{\overline{\left){\begin{align}
  & \overline{14}.\overline{00}\overline{00} \\
 & \underline{9} \\
 & 67\overline{\left){\begin{align}
  & 500 \\
 & \underline{469} \\
 & 744\overline{\left){\begin{align}
  & 3100 \\
 & \underline{2976} \\
 & \underline{124} \\
\end{align}}\right.} \\
\end{align}}\right.} \\
\end{align}}\right.}}\]
From the above long division, we got the required square root of the given number as \[3.74\].

Hence, we found the square root of $14$ to be equal to \[3.74\].

Note: Apart from using the long division method for calculating the square roots of the imperfect squares, we can also use the derivatives for this purpose. Also, several numerical methods like the Newton-Raphson method also exist for finding out these square roots.