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Square Root of Decimal Numbers

The square root will have half the number of decimal places as the number itself hasHence to calculate the square root of a decimal perfect square we should remember this:
 

Example :  since 0.1 x 0.1 = 0.01
                     since 0.8 x 0.8 = 0.64
In each case, the number has two digits in the decimal part while the square root has only one.

 

Example

 Find the square root of : 

(i) 0.0169  (ii) 0.422

 Solution

 1. The square root of 0.0169

  will have two decimal places. The square root of 169 is 13. Therefore the square root     of 0.0169 = 0.13.

  = 0.13
 

 2. Consider 4225. It ends with 25 and 6 x 7 = 42.
 Therefore, .  
 Square root of 0.4225 will have 2 decimal places.

 Therefore  .

 

Square Root of Decimals by the Method of Long Division

 For decimal numbers, pairing off in periods is done in the following way.

(i) For the whole number part before the decimal point, pairing is done by moving in pairs from right to left, as in the case of integers.
 

(ii) For the decimal part after the decimal point, pairing is done by moving in pairs from left to right. Also, the number of decimal places is first made even by putting a zero at the end, if necessary.

Thus 529.3845 is paired as


       53248.48659 is paired as

After pairing, the square root is found by long division method as explained earlier.
 
Example

Find the square root of 7624.7824.

Solution

 

Step 1
Pair off the numbers before the decimal point from right to left. The periods are 76 and 24.
 
Step 2
Since the number of decimal places is even, there is no need to put 0. Pair off the numbers after the decimal point from left to right. The periods are 78 to 24.
 
Step 3
Compute the square root by the long division method as shown.


 

Example

Find the square root of 0.00021025

Solution


There is an even number of decimal places. Mark off periods and calculate the square root by the long division method as shown.
 
Notice that the first period after the decimal is '00'.
Therefore a '0' is put in the quotient.

 
Example

Find the value of and hence find the sum of and .

Solution

 


   = 214

...  

(since 4.5796 has four decimal places, hence the square root will have two decimal places)
and (square root will have one decimal place)
...

 

Example

Estimate

Solution

100 < 101 < 144.

10 < < 12

So, the square root of 101 lies between 10 and 12

 

Example

Find and hence find the quotient of

Solution



 

Estimate the square roots:

Raja is playing in a square garden of area 61 square meters. Raja wants to know the length of the garden approximately.

How can we find the length of the side?

Let us calculate it.

Think of a perfect square number which is closest to 61 and greater than 61.

64.

Think of a perfect square number which is closest to 61 and less than 61.

49

49 < 61 < 64

7 <  < 8.

Since side of a square is square root of its area, we can conclude that the side of the square garden lies between 7m and 8m.





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