Estimate sqrt(150) Step by Step | Estimating square roots

Step by step

The question

Question: estimate √150.

√150

The square root of 150 is the number that times itself makes 150.

√150 × √150 = 150

Careful!

The square root is not half.

75 times 75 is 5625, far more than 150.

Careful: not √150 = 75 ✗

1 List the squares

First, the square numbers near 150.

11 times 11 is 121.

12 times 12 is 144.

13 times 13 is 169.

14 times 14 is 196.

2 Find where it fits

150 is between the squares 144 and 169.

= 144 < 150 < 169

A bigger number has a bigger square root.

So √150 is between √144 and √169.

= √144 < √150 < √169

√144 is 12.

√169 is 13.

So √150 is between 12 and 13.

= 12 < √150 < 13

3 Which is closer

150 is 6 more than 144, and 19 less than 169.

= 150 − 144 = 6, 169 − 150 = 19

6 is less.

So 150 is closer to 144, and √150 is closer to 12.

6 < 19: closer to 144

So √150 is between 12 and 13, closer to 12.

= 12 < √150 < 13, closer to 12

4 Check

Let's check with the number halfway: 12.5.

12.5 times 12.5 is 156.25, more than 150.

12.5 × 12.5 = 156.25 > 150

So √150 is below 12.5: closer to 12.

The answer is right!

√150 < 12.5: closer to 12

Let's look back.

First: list the squares.

Then: find where it fits.

Then: which is closer.

Last: check.

Answer