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.

We will illustrate this with a couple of 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.

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 quick.

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.

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