Estimate √10 Step by Step | Estimating square roots

Step by step

The question

Question: estimate √10.

√10

The square root of 10 is the number that times itself makes 10.

√10 × √10 = 10

Careful!

The square root is not half.

5 times 5 is 25, far more than 10.

Careful: not √10 = 5 ✗

1 List the squares

First, the square numbers near 10.

2 times 2 is 4.

3 times 3 is 9.

4 times 4 is 16.

5 times 5 is 25.

2 Find where it fits

10 is between the squares 9 and 16.

= 9 < 10 < 16

A bigger number has a bigger square root.

So √10 is between √9 and √16.

= √9 < √10 < √16

√9 is 3.

√16 is 4.

So √10 is between 3 and 4.

= 3 < √10 < 4

3 Which is closer

10 is 1 more than 9, and 6 less than 16.

= 10 − 9 = 1, 16 − 10 = 6

1 is less.

So 10 is closer to 9, and √10 is closer to 3.

1 < 6: closer to 9

So √10 is between 3 and 4, closer to 3.

= 3 < √10 < 4, closer to 3

4 Check

Let's check with the number halfway: 3.5.

3.5 times 3.5 is 12.25, more than 10.

3.5 × 3.5 = 12.25 > 10

So √10 is below 3.5: closer to 3.

The answer is right!

√10 < 3.5: closer to 3

Let's look back.

First: list the squares.

Then: find where it fits.

Then: which is closer.

Last: check.

Answer