Question

What is the square root of $26$?

Answer 1

Hint: The given number $26$ in the above question whose square root is to be determined is an imperfect square. Since the square root of an imperfect square is irrational, so will be the square root of the given number $26$. The approximate values of these types of irrational square roots are calculated by using the method of long division. Therefore, by using the long division method, we will get the approximate value of the square root of the given number.

Complete step by step solution:
According to the question, the number whose square root is to be determine is given to be $26$. The given number is not equal to the square of any of the natural numbers. Therefore, we can say that the given number is an imperfect square. We know that the square root of an imperfect square is an irrational number. We also know that the irrational numbers are those which have a decimal expansion which neither terminates nor repeats. So the decimal expansion for the square root of $26$ will also be non-terminating and non-repeating. Therefore, we cannot determine its exact values but only its approximate decimal expansion.
Now, we know that the square roots of the imperfect squares are obtained by using the long division method. Therefore, for calculating the square root of $26$, we use the long division method as shown below.
\[5\overset{5.09}{\overline{\left){\begin{align}
  & \overline{26}.\overline{00}\overline{00} \\
 & \underline{25} \\
 & 100\overline{\left){\begin{align}
  & 100 \\
 & \underline{0} \\
 & 1009\overline{\left){\begin{align}
  & 10000 \\
 & \underline{9081} \\
 & \underline{919} \\
\end{align}}\right.} \\
\end{align}}\right.} \\
\end{align}}\right.}}\]
From the above long division, we got the required square root equal to \[5.09\].

Hence, the square root of $26$ is equal to \[5.09\].

Note: We know that $26=2\times 13$. Therefore, using the law of radical given by $\sqrt{ab}=\sqrt{a}\sqrt{b}$, we can write the given square root as $\sqrt{2}\sqrt{13}$ and thus multiply the square roots of two and thirteen to get the square root of $26$. Since $26$ is just one less than $25$, which is equal to $5$, the square root of $26$ came close to $5$.