Estimate √120 Step by Step | Estimating square roots

Step by step

The question

Question: estimate √120.

√120

The square root of 120 is the number that times itself makes 120.

√120 × √120 = 120

Careful!

The square root is not half.

60 times 60 is 3600, far more than 120.

Careful: not √120 = 60 ✗

1 List the squares

First, the square numbers near 120.

9 times 9 is 81.

10 times 10 is 100.

11 times 11 is 121.

12 times 12 is 144.

2 Find where it fits

120 is between the squares 100 and 121.

= 100 < 120 < 121

A bigger number has a bigger square root.

So √120 is between √100 and √121.

= √100 < √120 < √121

√100 is 10.

√121 is 11.

So √120 is between 10 and 11.

= 10 < √120 < 11

3 Which is closer

120 is 20 more than 100, and 1 less than 121.

= 120 − 100 = 20, 121 − 120 = 1

1 is less.

So 120 is closer to 121, and √120 is closer to 11.

1 < 20: closer to 121

So √120 is between 10 and 11, closer to 11.

= 10 < √120 < 11, closer to 11

4 Check

Let's check with the number halfway: 10.5.

10.5 times 10.5 is 110.25, less than 120.

10.5 × 10.5 = 110.25 < 120

So √120 is above 10.5: closer to 11.

The answer is right!

√120 > 10.5: closer to 11

Let's look back.

First: list the squares.

Then: find where it fits.

Then: which is closer.

Last: check.

Answer