Estimate √390 Step by Step | Estimating square roots

Step by step

The question

Question: estimate √390.

√390

The square root of 390 is the number that times itself makes 390.

√390 × √390 = 390

Careful!

The square root is not half.

195 times 195 is 38025, far more than 390.

Careful: not √390 = 195 ✗

1 List the squares

First, the square numbers near 390.

18 times 18 is 324.

19 times 19 is 361.

20 times 20 is 400.

21 times 21 is 441.

2 Find where it fits

390 is between the squares 361 and 400.

= 361 < 390 < 400

A bigger number has a bigger square root.

So √390 is between √361 and √400.

= √361 < √390 < √400

√361 is 19.

√400 is 20.

So √390 is between 19 and 20.

= 19 < √390 < 20

3 Which is closer

390 is 29 more than 361, and 10 less than 400.

= 390 − 361 = 29, 400 − 390 = 10

10 is less.

So 390 is closer to 400, and √390 is closer to 20.

10 < 29: closer to 400

So √390 is between 19 and 20, closer to 20.

= 19 < √390 < 20, closer to 20

4 Check

Let's check with the number halfway: 19.5.

19.5 times 19.5 is 380.25, less than 390.

19.5 × 19.5 = 380.25 < 390

So √390 is above 19.5: closer to 20.

The answer is right!

√390 > 19.5: closer to 20

Let's look back.

First: list the squares.

Then: find where it fits.

Then: which is closer.

Last: check.

Answer