Estimate the square root

One way to estimate the square root of any number is to find a whole number greater than the square root and another whole number less than the square root. 

Take a close look at the figure below so you can learn the process.

There is a very important observation you need to make!

In our example above, notice that 34 is much closer to 36 than it is to 25.

Therefore, we chose a number very close to 6 and that number is 5.8.

Estimate the square root

More examples showing how to estimate the square root

We will illustrate the process with two more examples.

Example #1:

Estimate the square root of 17

We will find a whole number bigger than the square root of 17 and a whole number smaller than the square root of 17. 

<   17   <


Study carefully the procedure!

First, let us find the number that is bigger than   17


17   <   
18

17   <   
19

17   <   
20

17   <   
21

17   <   
22

17   <   
23

17   <   
24

17   <   
25


Notice that the square roots of 18, 19, 20, 21, 22, 23, 24 are all bigger than the square root of 17.

However, only   25   will give a whole number, so this is the one we will choose.


25 = 5 since 5 × 5 = 25


Second, let us find the number that is smaller than   17


16   <   
17


Since   16   is a whole number, this is the one we will choose.


16   = 4 since 4 × 4 = 16


We get 4   < 17   < 5


The square root of 17 is between 4 and 5. We could estimate the square of 17 to be 4.1 for example.

Example #2:

Estimate the square root of 102

We will find a whole number bigger than the square root of 102 and a whole number smaller than the square root of 102. 

101   <   
102


100   <   
102


We will use square root of 100 since   100   = 10


To find the number bigger than the square of 102, we will use a different strategy. It take too long to write down the square root of all these numbers. Some good observation will help us to solve the problem quickly.

Notice that the square root of any number between 103 and 120 is not a whole number.

However, square root of 121 is a whole number since 11 times 11 = 121.

Therefore, the square of 121 will give us the whole number that we need that is bigger than square of 102.



We get 10   < 102   < 11


The square root of 102 is between 10 and 11. We could estimate the square root of 102 to be 10.2.

Notice again that in our estimation, we chose a number close to 10 since 102 is much closer 100 than it is 121.

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