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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