## Finding Square root of Perfect Square Decimal Numbers by Division Method

A slight variation in method is necessary when it is required to find the square root of a decimal.
While finding the square root of a natural number, say, 46656, you make pairs by counting from right to left and if in the last, one digit is left, you leave it by itself. For example, while finding the square root of 276676 and 46656, we form pairs as under:

$\leftarrow$  $\leftarrow$  $\leftarrow$
27 66 76
$\leftarrow$  $\leftarrow$  $\leftarrow$
4  66  56

In case of a decimal number, we count from left to the right for the decimal portion and from right to the left for the integral portion.
If the last period of the decimal number contains only one figure we may add zero to it. This is because two digits are necessary to make up a period, while the addition of a zero at the right of a decimal figure does not change its value. For example, while finding the square roots of 0.00002601, 998.56 the periods will be formed as under:

$\rightarrow$  $\rightarrow$  $\rightarrow$ $\rightarrow$       $\leftarrow$  $\rightarrow$
0.00 00 26 01, 4 92.84

The square root of a decimal number will contain as many decimal places as there are periods, or half as many decimal places as the given number.
The operations in obtaining the square root of a decimal number are the same as for whole numbers.
Follow the steps in the following example:

Example 1: Find the square root of 3881.29.
Solution:

Step 1. Beginning at the decimal point, mark off points to the left and right.

Step 2. 6 is the largest whole-number square root that is contained in 38 which constitutes the first period. Write 6 on top as quotient and also in the divisor. Subtract 6 x 6 = 36 from 38. The remainder is 2.

Step 3. Bring down the next pair 81. Double 6 (the number in the quotient) and place the double, i.e., 12 as the next divisor as shown. Divide 12 into 28 to obtain 2. Write 2 as the next number in the quotient and also place 2 next to 12 in the divisor as shown. Multiply 122 by 2 and place the product under 281. Subtract. The remainder is 37.

Step 4. Place the decimal point in the quotient after 2 because the next pair is a decimal part.

Step 5. Bring down 29 to make 3729 the next dividend. Double 62 (the number in the quotient) and place the double, i.e., 124 as the next divisor as shown. Divide 124 into 372 to obtain 3. Write 3 as the next number in the quotient and also place 3 next to 124 in the divisor. Multiply 1243 by 3 and place the product under 3729.
Subtract. The remainder is zero.

$\therefore$     $\sqrt { 3881.29 }$ = 62.3

Example 2: Find the square root of 6432.04.
Solution : In the above example, since 16 does not divide 3, put a zero both in the root (quotient) and the divisor and bring down the next pair 04 also.

Example 3: Find the value of sf3136 and use it to find the value of $\sqrt { 31.36 }$ + $\sqrt { 0.3136 }$
Solution : $\sqrt { 31.36 }$ = 5.6

$\therefore$  $\sqrt { 31.36 }$ = 5.6 and  $\sqrt { 0.3136 }$ = 0.56
$\therefore$  $\sqrt { 31.36 }$ + $\sqrt { 0.3136 }$ = 5.6 + 0.56 = 6.16

We can also work out as under
$\sqrt { 31.36 }$ + $\sqrt { 0.3136 }$

$\sqrt { \frac { 3136 } { 100 } }$ + $\sqrt { \frac { 3136 } { 10000 } }$

$\frac { \sqrt { 3136 } } { 10 }$  + $\frac { \sqrt { 3136 } } { 100 }$

=   $\frac {56 }{ 10 }$$\frac { 56 }{ 100 }$ = 5.6 + 0.56 = 6.16

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