Question

What is the square root for 40.5?

Answer 1

Hint: To find the square root for 40.5, we have to combine the number 40.5 in pairs such that we have to put a bar on each pair starting from the unit place for the integral part. Similarly, we have to put a bar on the pairs after the decimal. Hence, we have to rewrite 40.5 as 40.50 and put a bar like $\overline{40}.\overline{50}$ . We then have to divide this number with a divisor which will be the maximum number whose square is less than or equal to the dividend. Then write the remainder and bring down the next pair of the dividend so that we will get the new dividend. Now, we have to find the new divisor by summing the old divisor and the digit in the unit place of the odd divisor. We will leave the unit place after this result blank. We have to choose the unit place of the divisor such that it will be the largest digit which when multiplied with the new divisor will be equal to or less than the new dividend. We will perform the division process till the remainder becomes zero or till two or three decimal places.

Complete step-by-step solution:
We have to find the square root for 40.5. We have to combine the number 40.5 in pairs. So we have to rewrite 40.5 as 40.50. Now, we have to put a bar on each pair starting from the unit place for the integral part. Similarly, we have to put a bar on the pairs after the decimal. Hence, we will denote this as $\overline{40}.\overline{50}$ .
Now, we have to divide $\overline{40}.\overline{50}$ . The divisor will be the maximum number whose square is less than or equal to the dividend (that is, the integral part of the number). Here, we will consider the dividend as 40. Now, we have to find the divisor. We know that ${{6}^{2}}=36<40$ and ${{7}^{2}}=49>40$ . Therefore, we will consider the divisor to be 6. Now, we have to perform the division process and write the remainder.
\[\begin{align}
  & \begin{matrix}
   {} & 6 \\
\end{matrix} \\
 & \begin{matrix}
   6 \\
   {} \\
   {} \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \overline{40}.\text{ }\overline{50} \\
 & -36 \\
 & \_\_\_\_\_\_\_\_ \\
 & \begin{matrix}
   4 & {} \\
\end{matrix} \\
\end{align} \,}} \right. \\
\end{align}\]
Now, we have to take the next pair below. Now, the new dividend will be 450. The new divisor will be the sum of the old divisor and the digit in the unit place of the odd divisor. Hence the new divisor can be written as 12_.
\[\begin{align}
  & \begin{matrix}
   {} & \text{ } \text{ } \text{ } \text{ } \text{ } \text{ }6 \\
\end{matrix} \\
 & \begin{matrix}
   6 \\
   {} \\
   12\_ \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \overline{40}.\text{ }\overline{50} \\
 & -36 \\
 & \_\_\_\_\_\_\_\_ \\
 & \begin{matrix}
   4 & 50 \\
\end{matrix} \\
\end{align} \,}} \right. \\
\end{align}\]
We have to choose the unit place of the divisor such that it will be the largest digit which when multiplied with the new divisor will be equal to or less than the dividend (450). We can see that $123\times 3=369<450$ and $124\times 4=496>450$ . Therefore, the new dividend will be 123. Now, we will perform the division. The quotient will be written after the decimal point since 50 is after the decimal point.
\[\begin{align}
  & \begin{matrix}
   {} & \begin{matrix}
   {} & 6.3 & {} \\
\end{matrix} \\
\end{matrix} \\
 & \begin{matrix}
   6 \\
   {} \\
   123 \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \overline{40}.\text{ }\begin{matrix}
   \overline{50} & \overline{00} \\
\end{matrix} \\
 & -36 \\
 & \_\_\_\_\_\_\_\_\_\_ \\
 & \begin{matrix}
   4 & \text{ }50 \\
\end{matrix} \\
 & \begin{matrix}
   -3 & 69 \\
\end{matrix} \\
 & \_\_\_\_\_\_\_\_\_\_\_ \\
 & \begin{matrix}
   {} & \text{ } \text{ } \text{ }\text{ }\begin{matrix}
   81 & 00 \\
\end{matrix} \\
\end{matrix} \\
\end{align} \,}} \right. \\
\end{align}\]
We will add a new pair of zeroes since the remainder is not zero. Now, the new dividend is 8100. The divisor will be $123+3=126$ , that is, 246_. We have to find the unit digit. $1266\times 6=7596<8100$ .
\[\begin{align}
  & \begin{matrix}
   {} & \begin{matrix}
   {} &\text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } 6. & \begin{matrix}
   3 & 6 \\
\end{matrix} \\
\end{matrix} \\
\end{matrix} \\
 & \begin{matrix}
   6 \\
   \begin{matrix}
   \begin{matrix}
   \begin{matrix}
   \begin{matrix}
   {} \\
   123 \\
   {} \\
\end{matrix} \\
   {} \\
\end{matrix} \\
   1266 \\
\end{matrix} \\
   {} \\
\end{matrix} \\
   \begin{matrix}
   {} \\
   {} \\
   {} \\
\end{matrix} \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \overline{40}.\text{ }\begin{matrix}
   \overline{50} & \overline{00} \\
\end{matrix} \\
 & -36 \\
 & \_\_\_\_\_\_\_\_\_\_ \\
 & \begin{matrix}
   4 & \text{ }50 \\
\end{matrix} \\
 & \begin{matrix}
   -3 & 69 \\
\end{matrix} \\
 & \_\_\_\_\_\_\_\_\_\_\_ \\
 & \begin{matrix}
   {} & \begin{matrix}
   {} & \begin{matrix}
   81 & 00 \\
\end{matrix} \\
\end{matrix} \\
\end{matrix} \\
 & \begin{matrix}
   {} & \begin{matrix}
   {} & 75 \\
\end{matrix} & 96 \\
\end{matrix} \\
 & \_\_\_\_\_\_\_\_\_\_\_\_ \\
 & \begin{matrix}
   {} & \begin{matrix}
   {} & 5 & 04 \\
\end{matrix} \\
\end{matrix} \\
\end{align} \,}} \right. \\
\end{align}\]
We will evaluate till 3 decimal places only.
Hence, the square root 40.5 is 6.36.

Note: We can also find the square root of 40.5 in an alternate way. Let us write 40.5 in fractional form. For this, we have to multiply and divide 40.5 by 10 since only one digit is there after the decimal.
$\Rightarrow 40.5=\dfrac{40.5\times 10}{1\times 10}=\dfrac{405}{10}$
Let us cancel the common factor 5.
$\Rightarrow \dfrac{405}{10}=\dfrac{81}{2}$
Now, we have to find $\sqrt{40.5}$ .
$\Rightarrow \sqrt{40.5}=\sqrt{\dfrac{81}{2}}$
We know that $\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}$ .
$\Rightarrow \sqrt{\dfrac{81}{2}}=\dfrac{\sqrt{81}}{\sqrt{2}}=\dfrac{9}{\sqrt{2}}$
Let us rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$ .
$\Rightarrow \dfrac{9\times \sqrt{2}}{\sqrt{2}\times \sqrt{2}}=\dfrac{9\sqrt{2}}{2}$
We can either leave the answer as it is or convert it into decimals. We know that $\sqrt{2}=1.414$ .
$\Rightarrow \dfrac{9\sqrt{2}}{2}=\dfrac{9\times 1.414}{2}=6.363$
Therefore $\sqrt{40.5}=\dfrac{9\sqrt{2}}{2}=6.363$ .