Estimate √10 to the nearest tenth Step by Step

Step by step

The question

Question: estimate √10 to the nearest tenth.

√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

4 Try tenths

√10 is a bit more than 3.

So we try 3.1, 3.2 and so on, and square each one.

3.1 times 3.1 is 9.61.

That is less than 10: too small.

= 3.1 × 3.1 = 9.61 → too small

3.2 times 3.2 is 10.24.

That is more than 10: too big.

= 3.2 × 3.2 = 10.24 → too big

So √10 is between 3.1 and 3.2.

= 3.1 < √10 < 3.2

5 Pick the nearest tenth

10 is 0.39 more than 9.61, and 0.24 less than 10.24.

So it is closer to 10.24.

= 10 − 9.61 = 0.39, 10.24 − 10 = 0.24

0.24 < 0.39: closer to 3.2

So √10 is about 3.2, to the nearest tenth.

= √10 ≈ 3.2

6 Check

Let's check with the number halfway: 3.15.

3.15 times 3.15 is 9.9225, less than 10.

3.15 × 3.15 = 9.9225 < 10

So √10 is above 3.15.

To the nearest tenth, it is 3.2.

The answer is right!

√10 > 3.15: nearest tenth 3.2

Let's look back.

First: list the squares.

Then: find where it fits.

Then: which is closer.

Then: try tenths.

Then: pick the nearest tenth.

Last: check.

Answer