Find the square root using the long division method: $\sqrt {101\dfrac{1}{{400}}} $.

Hint: First, convert the mixed fraction into an improper fraction. Then find the square root of the numerator by long division method. After that find the square root of the denominator by long division method. After that put the value in fraction. Now, convert the improper fraction in the mixed fraction to get the solution.

Complete step-by-step answer:
Given:- $\sqrt {101\dfrac{1}{{400}}} $
For an improper fraction of $101\dfrac{1}{{400}}$, multiply 101 with 400 and add 1 to it.
So, the value in an improper fraction is $\dfrac{{40401}}{{400}}$.
Now, find the square root of 40401 by long division method.
Separate the digits by taking bars from right to left after two digits.
$\overline 4 \,\overline {04} \,\overline {01} $
Now, multiply a number by itself such that the product must be less than or equal to 4. The condition will be met by “2” and the remainder will be 0. Then,
$
  \left. 2 \right)\overline 4 \overline {04} \overline {01} \left( 2 \right. \\
  \,\,\,\,\,\underline 4 \\
  \,\,\,\,\,0 \\
 $
Now, bring down 04. The new dividend is 4. Now double the quotient to get a new divisor. Also, concatenate it with a suitable digit such that the product of the new divisor and the digit is less than or equal to the new dividend.
$
  \left. {\,\,\,2} \right)\overline 4 \overline {04} \overline {01} \left( {20} \right. \\
  \,\,\,\,\,\,\,\,\underline 4 \\
  \left. {40} \right)004 \\
  \,\,\,\,\,\,\,\,\,\,\underline {00} \\
  \,\,\,\,\,\,\,\,\,\,04 \\
 $
Now, bring down 01. The new dividend is 401. Now again double the quotient to get a new divisor. Also, concatenate it with a suitable digit such that the product of the new divisor and the digit is less than or equal to the new dividend.
\[
  \,\,\,\,\,\left. 2 \right)\overline 4 \overline {04} \overline {01} \left( {201} \right. \\
  \,\,\,\,\,\,\,\,\,\,\underline 4 \\
  \,\,\left. {40} \right)004 \\
  \,\,\,\,\,\,\,\,\,\,\underline {00} \\
  \,\,\,\,\,\,\,\,\,\,\,\,401 \\
  \left. {401} \right)\,\,\,401 \\
 \]
Thus, the square root of the number 40401 is 201.
Now, find the square root of 400 by long division method.
Separate the digits by taking bars from right to left after two digits.
$\overline 4 \,\overline {00} $
Now, multiply a number by itself such that the product must be less than or equal to 4. The condition will be met by “2” and the remainder will be 0. Then,
$
  \left. 2 \right)\overline 4 \overline {00} \left( 2 \right. \\
  \,\,\,\,\,\underline 4 \\
  \,\,\,\,\,0 \\
 $
Now, bring down 04. The new dividend is 4. Now double the quotient to get a new divisor. Also, concatenate it with a suitable digit such that the product of the new divisor and the digit is less than or equal to the new dividend.
$
  \left. {\,\,\,2} \right)\overline 4 \overline {00} \left( {20} \right. \\
  \,\,\,\,\,\,\,\,\underline 4 \\
  \left. {40} \right)000 \\
  \,\,\,\,\,\,\,\,\,\,\underline {00} \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,0 \\
 $
Thus, the square root of the number 400 is 20.
So the square root of $\dfrac{{40401}}{{400}}$ is $\dfrac{{201}}{{20}}$.
Now, $\dfrac{{201}}{{20}}$ in a mixed fraction is $10\dfrac{1}{{20}}$.

Hence, $\sqrt {101\dfrac{1}{{400}}} = 10\dfrac{1}{{20}}$.

Note: Steps of Long Division Method for Finding Square Roots:
First, group the digits in pairs. Start with the digit in the units place.
Take the largest number whose square is equal to or just less than the first period. Then, take the number as the divisor and also as the quotient.
Subtract the product of the divisor and the quotient from the first period and bring down the next period to the right of the remainder. This becomes the new dividend.
Now, the new divisor is obtained by taking two times the quotient and concatenating with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend.
Repeat the steps (2), (3), and (4) till all the periods have been taken up. Now, the quotient so obtained is the required square root of the given number.