Estimate √300 Step by Step | Estimating square roots

Step by step

The question

Question: estimate √300.

√300

The square root of 300 is the number that times itself makes 300.

√300 × √300 = 300

Careful!

The square root is not half.

150 times 150 is 22500, far more than 300.

Careful: not √300 = 150 ✗

1 List the squares

First, the square numbers near 300.

16 times 16 is 256.

17 times 17 is 289.

18 times 18 is 324.

19 times 19 is 361.

2 Find where it fits

300 is between the squares 289 and 324.

= 289 < 300 < 324

A bigger number has a bigger square root.

So √300 is between √289 and √324.

= √289 < √300 < √324

√289 is 17.

√324 is 18.

So √300 is between 17 and 18.

= 17 < √300 < 18

3 Which is closer

300 is 11 more than 289, and 24 less than 324.

= 300 − 289 = 11, 324 − 300 = 24

11 is less.

So 300 is closer to 289, and √300 is closer to 17.

11 < 24: closer to 289

So √300 is between 17 and 18, closer to 17.

= 17 < √300 < 18, closer to 17

4 Check

Let's check with the number halfway: 17.5.

17.5 times 17.5 is 306.25, more than 300.

17.5 × 17.5 = 306.25 > 300

So √300 is below 17.5: closer to 17.

The answer is right!

√300 < 17.5: closer to 17

Let's look back.

First: list the squares.

Then: find where it fits.

Then: which is closer.

Last: check.

Answer