Question

What is the square root of 90?

Answer 1

Hint: To find the square root of 90, we will use a long division method. We have to place a bar on top of each pair of digits of the given number starting from the unit place. If we have an odd number of places, then the first digit will have a bar. We will then take the divisor which will be the largest number whose square is less than or equal to the number on the. Then we will divide and write the quotient and remainder. We will also extend zeroes in pairs to the right since 90 is not a perfect square. Then, we will bring down the next pair in the dividend. We will double the quotient and write it on the left as a divisor with a blank space. This unit place will be filled by a number which is the largest such that when we multiply the whole number after filling the unit place with the number in the unit place, we must get the result that is equal to or less than the new dividend. We will then repeat this procedure.

Complete step by step solution:
We have to find the square root of 90. Let us use long division method for this. Firstly, we have to place a bar on top of each pair of digits of the given number starting from the unit place. If we have odd number of places, then the first digit will have a bar. We will take the divisor which will be the largest number whose square is less than or equal to the number on the. Here, we can take 9 as the divisor, since ${{9}^{2}}=81<\text{Number on the left}$ . We have to divide 90 by 9 and write its quotient. We will also extend zeroes in pairs to the right since 90 is not a perfect square.
\[\begin{align}
  & \begin{matrix}
   \text{ } & 9 & {} \\
\end{matrix} \\
 & \begin{matrix}
   9 \\
   {} \\
   {} \\
   {} \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \overline{90}.\overline{00} \\
 & -81 \\
 & \_\_\_\_\_ \\
 & \begin{matrix}
   {} & 9 \\
\end{matrix} \\
\end{align} \,}} \right. \\
\end{align}\]
Now, we will bring down the next pair in the dividend, that is, 00. We will double the quotient and write it on the left as divisor with a blank space. Here, we will perform $9\times 2=18$ .
\[\begin{align}
  & \begin{matrix}
   \text{ } & {} & 9 \\
\end{matrix} \\
 & \begin{matrix}
   9 \\
   {} \\
   {} \\
   18\_ \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \overline{90}.\overline{00} \\
 & -81 \\
 & \_\_\_\_\_ \\
 & \begin{matrix}
   {} & 900 \\
\end{matrix} \\
\end{align} \,}} \right. \\
\end{align}\]
Now, for the unit place of the divisor, we have to choose a number which is the largest such that when we multiply the whole number after filling the unit place with the number in the unit place, we must get the result that is equal to or less than the dividend (900). Here, we will take 4, so that $184\times 4=736$ . Then, we will perform the division.
\[\begin{align}
  & \begin{matrix}
   \text{ } & {} & 94 \\
\end{matrix} \\
 & \begin{matrix}
   9 \\
   {} \\
   184 \\
   {} \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \overline{90}.\overline{00} \\
 & -81 \\
 & \_\_\_\_\_ \\
 & \begin{matrix}
   {} & 900 \\
\end{matrix} \\
 & \begin{matrix}
   {} & -736 \\
\end{matrix} \\
 & \_\_\_\_\_\_\_\_ \\
 & \begin{matrix}
   {} & 164 \\
\end{matrix} \\
\end{align} \,}} \right. \\
\end{align}\]
We will now double 94 and write it as the divisor. We will also extend the zeroes to the dividend.
\[\begin{align}
  & \begin{matrix}
   \text{ } & {} & 94 \\
\end{matrix} \\
 & \begin{matrix}
   9 \\
   {} \\
   184 \\
   \begin{matrix}
   {} \\
   {} \\
   188\_ \\
\end{matrix} \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \overline{90}.\overline{00}\overline{00} \\
 & -81 \\
 & \_\_\_\_\_ \\
 & \begin{matrix}
   {} & 900 \\
\end{matrix} \\
 & \begin{matrix}
   {} & -736 \\
\end{matrix} \\
 & \_\_\_\_\_\_\_\_ \\
 & \begin{matrix}
   {} & 16400 \\
\end{matrix} \\
\end{align} \,}} \right. \\
\end{align}\]
The unit place of the divisor will be 8 so that $1888\times 8=15104<16400$ .
\[\begin{align}
  & \begin{matrix}
   \text{ } & {} & 9.4 \\
\end{matrix}8 \\
 & \begin{matrix}
   9 \\
   {} \\
   184 \\
   \begin{matrix}
   {} \\
   {} \\
   1888 \\
\end{matrix} \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \overline{90}.\overline{00}\overline{00} \\
 & -81 \\
 & \_\_\_\_\_ \\
 & \begin{matrix}
   {} & 900 \\
\end{matrix} \\
 & \begin{matrix}
   - & 736 \\
\end{matrix} \\
 & \_\_\_\_\_\_\_\_ \\
 & \begin{matrix}
   {} & 16400 \\
\end{matrix} \\
 & \begin{matrix}
   - & 15104 \\
\end{matrix} \\
 & \_\_\_\_\_\_\_\_\_\_ \\
 & \begin{matrix}
   {} & 1296 \\
\end{matrix} \\
\end{align} \,}} \right. \\
\end{align}\]
Hence, the root of 90 is 9.48.

Note: We can also find the square root of 90 using the prime factorization.
\[\begin{align}
  & 2\left| \!{\underline {\,
  90 \,}} \right. \\
 & 5\left| \!{\underline {\,
  45 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ 1} \\
\end{align}\]
We can write 90 as $2\times 5\times 3\times 3$ . We have to combine pairs of the same number. The different numbers will be inside the square root.
$\sqrt{90}=3\sqrt{2\times 5}=3\sqrt{10}$