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Find the square root of the following decimal numbers.
(i) 2.56 (ii) 18.49 (iii) 68.89 (iv) 84.64

Answer
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Hint:Here we will use the long division method in each of the parts to find the square root.Long division method is used to divide a large number (usually three digits or more) by a number having two or more digits.

Complete step-by-step answer:
Steps to find square root by long division method:
1. Place a bar over the pair of numbers starting from the unit place or Right-hand side of the number.
2. Take the largest number as the divisor whose square is less than or equal to the number on the extreme left of the number. The digit on the extreme left is the dividend. Divide and write the quotient.
3. Now, we then bring down the number, which is under the bar, to the right side of the remainder
4. Now double the value of the quotient and enter it with blank space on the right side. Next, we have to select the largest digit for the unit place of the divisor such that the new number, when multiplied by the new digit at unit’s place, is equal to or less than the dividend.
5. Continue the process till the remainder is zero and then write the quotient as the answer.

(i) The given decimal number is: 2.56
Now we will apply a long division method to find its square root.
Applying long division method we get:-
First we will place a bar over the pair of numbers starting from the unit place and also the decimals.
\[ \Rightarrow \overline 2 .\overline {56} \]
Now we will take the largest number as the divisor whose square is less than or equal to 2 then divide and write the quotient i.e. 1 and put a decimal point in the quotient as there is a decimal in the dividend.
\[
  1\mathop{\left){\vphantom{1{\overline 2 .\overline {56} }}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{\overline 2 .\overline {56} }}}}
\limits^{\displaystyle \,\,\, {1.}} \\
   - 1 \\
   \cdots \cdots \cdots \cdots \\
  {\text{ }}1 \\
   \cdots \cdots \cdots \cdots \\
 \]
Now we will bring down 56, which is under the bar, to the right side of the remainder and double the value of the quotient and write it with blank space on the right side. Next, we have to select the largest digit for the unit place of the divisor such that the new number, when multiplied by the new digit at unit’s place, is equal to or less than 156.
\[
  26\mathop{\left){\vphantom{1{156}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{156}}}}
\limits^{\displaystyle \,\,\, 6} \\
  {\text{ }} - 156 \\
   \cdots \cdots \cdots \cdots \cdots \\
  {\text{ 0}} \\
   \cdots \cdots \cdots \cdots \\
 \]
Now since we got the remainder as zero hence we can write the final quotient.
Hence the final quotient is 1.6.
Therefore the square root of 2.56 is 1.6.

(ii) The given decimal number is: 18.49
Now we will apply a long division method to find its square root.
Applying long division method we get:-
First we will place a bar over the pair of numbers starting from the unit place and also the decimals.
\[ \Rightarrow \overline {18} .\overline {49} \]
Now we will take the largest number as the divisor whose square is less than or equal to 18 then divide and write the quotient i.e. 4 and put a decimal point in the quotient as there is a decimal in the dividend.
\[
  4\mathop{\left){\vphantom{1{\overline {18} .\overline {49} }}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{\overline {18} .\overline {49} }}}}
\limits^{\displaystyle \,\,\, {4.}} \\
  {\text{ }} - 16 \\
   \cdots \cdots \cdots \cdots \\
  {\text{ 2}} \\
   \cdots \cdots \cdots \cdots \\
 \]
Now we will bring down 49, which is under the bar, to the right side of the remainder and double the value of the quotient and write it with blank space on the right side. Next, we have to select the largest digit for the unit place of the divisor such that the new number, when multiplied by the new digit at unit’s place, is equal to or less than 249.
\[
  83\mathop{\left){\vphantom{1{249}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{249}}}}
\limits^{\displaystyle \,\,\, 3} \\
  {\text{ }} - 249 \\
   \cdots \cdots \cdots \cdots \cdots \\
  {\text{ 0}} \\
   \cdots \cdots \cdots \cdots \\
 \]
Now since we got the remainder as zero hence we can write the final quotient.
Hence the final quotient is 4.3.
Therefore the square root of 18.49 is 4.3.

(iii) The given decimal number is: 68.89
Now we will apply a long division method to find its square root.
Applying long division method we get:-
First we will place a bar over the pair of numbers starting from the unit place and also the decimals.
\[ \Rightarrow \overline {68} .\overline {89} \]
Now we will take the largest number as the divisor whose square is less than or equal to 68 then divide and write the quotient i.e. 8 and put a decimal point in the quotient as there is a decimal in the dividend.
\[
  8\mathop{\left){\vphantom{1{\overline {68} .\overline {89} }}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{\overline {68} .\overline {89} }}}}
\limits^{\displaystyle \,\,\, {8.}} \\
  {\text{ }} - 64 \\
   \cdots \cdots \cdots \cdots \\
  {\text{ 4}} \\
   \cdots \cdots \cdots \cdots \\
 \]
Now we will bring down 89, which is under the bar, to the right side of the remainder and double the value of the quotient and write it with blank space on the right side. Next, we have to select the largest digit for the unit place of the divisor such that the new number, when multiplied by the new digit at unit’s place, is equal to or less than 489.
\[
  163\mathop{\left){\vphantom{1{489}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{489}}}}
\limits^{\displaystyle \,\,\, 3} \\
  {\text{ }} - 489 \\
   \cdots \cdots \cdots \cdots \cdots \\
  {\text{ 0}} \\
   \cdots \cdots \cdots \cdots \\
 \]
Now since we got the remainder as zero hence we can write the final quotient.
Hence the final quotient is 8.3.
Therefore the square root of 68.89 is 8.3.

(iv) The given decimal number is: 84.64
Now we will apply a long division method to find its square root.
Applying long division method we get:-
First we will place a bar over the pair of numbers starting from the unit place and also the decimals.
\[ \Rightarrow \overline {84} .\overline {64} \]
Now we will take the largest number as the divisor whose square is less than or equal to 84 then divide and write the quotient i.e. 9 and put a decimal point in the quotient as there is a decimal in the dividend.
\[
  9\mathop{\left){\vphantom{1{\overline {84} .\overline {64} }}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{\overline {84} .\overline {64} }}}}
\limits^{\displaystyle \,\,\, {9.}} \\
  {\text{ }} - 81 \\
   \cdots \cdots \cdots \cdots \\
  {\text{ 3}} \\
   \cdots \cdots \cdots \cdots \\
 \]
Now we will bring down 64, which is under the bar, to the right side of the remainder and double the value of the quotient and write it with blank space on the right side. Next, we have to select the largest digit for the unit place of the divisor such that the new number, when multiplied by the new digit at unit’s place, is equal to or less than 364.
\[
  182\mathop{\left){\vphantom{1{364}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{364}}}}
\limits^{\displaystyle \,\,\, 2} \\
  {\text{ }} - 364 \\
   \cdots \cdots \cdots \cdots \cdots \\
  {\text{ 0}} \\
   \cdots \cdots \cdots \cdots \\
 \]
Now since we got the remainder as zero hence we can write the final quotient.
Hence the final quotient is 9.2.
Therefore the square root of 84.64 is 9.2.

Note:The student may make mistakes while selecting the right quotient, so one should follow the steps of the long division method carefully and should continue the process until the remainder comes out to be zero.