# ESTIMATE SQUARE ROOTS

When we estimate the square root of a number, first we will find the two values between which the square root of the given number lies.

For example square root of the number 40 lies between 6 and 7.

Because square root of 36 is 6 and square root of 49 is 7. The given number lies between 36 and 49. So that we can say that the square root of the given number is between 6 and 7.

√1  =  1

√4  =  2

√9  =  3

√16  =  4

√25  =  5

√36  =  6

√49  =  7

√64  =  8

√81  =  9

√100  =  10

√121  =  11

√144  =  12

√169  =  13

√196  =  14

√225  =  15

√256  =  16

√289  =  17

√324  =  18

√361  =  19

√400  =  20

√441  =  21

√484  =  22

√529  =  23

√576  =  24

√625  =  25

√676  =  26

√729  =  27

√784  =  28

√841  =  29

√900  =  30

From the above square roots, we can come to know that

(i) If a perfect square has ‘n’ digits where n is even, its square root has n/2 digits.

(ii) If a perfect square has ‘n’ digits where n is odd, its square root has (n + 1)/2 digits.

Example 1 :

Estimate the value of the following to the nearest whole number.

√80

Solution :

√64 < √80 < √81

8 < √80 < 9

Since the given number lies between 64 and 81, the approximate square root of the given number is 8.

So, the approximate value of √80 is 8.

Example 2 :

Estimate the value of the following to the nearest whole number.

√1000

Solution :

The given number is greater than  900. If we multiply 32 x 32 we will get 1024 which is greater than 1000. So we have to try the number less than 32 but greater than 30.

31 x 31  =  961 < 1000

So, the approximate value of  1000 is 31.

Example 3 :

Estimate the value of the following to the nearest whole number.

√172

Solution :

The given number is greater than 169 and lesser than 196. So the square root of the given number lies between 16 and 17.

So, the approximate value of 172 is 16.

Example 4 :

Estimate the value of the following to the nearest whole number.

√5928

Solution :

Number of digits in the given number = 4

So, square root of 5928 will have 2 digits.

70 x 70  =  4900

75 x 75  =  5625

77 x 77  =  5929

So, the approximate value 5928 is 76.

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