Estimate √65 to the nearest tenth Step by Step

Step by step

The question

Question: estimate √65 to the nearest tenth.

√65

The square root of 65 is the number that times itself makes 65.

√65 × √65 = 65

Careful!

The square root is not half.

32.5 times 32.5 is 1056.25, far more than 65.

Careful: not √65 = 32.5 ✗

1 List the squares

First, the square numbers near 65.

7 times 7 is 49.

8 times 8 is 64.

9 times 9 is 81.

10 times 10 is 100.

2 Find where it fits

65 is between the squares 64 and 81.

= 64 < 65 < 81

A bigger number has a bigger square root.

So √65 is between √64 and √81.

= √64 < √65 < √81

√64 is 8.

√81 is 9.

So √65 is between 8 and 9.

= 8 < √65 < 9

3 Which is closer

65 is 1 more than 64, and 16 less than 81.

= 65 − 64 = 1, 81 − 65 = 16

1 is less.

So 65 is closer to 64, and √65 is closer to 8.

1 < 16: closer to 64

4 Try tenths

√65 is a bit more than 8.

So we try 8.1 and so on, and square each one.

8.1 times 8.1 is 65.61.

That is more than 65: too big.

= 8.1 × 8.1 = 65.61 → too big

So √65 is between 8.0 and 8.1.

= 8.0 < √65 < 8.1

5 Pick the nearest tenth

65 is 1 more than 64, and 0.61 less than 65.61.

So it is closer to 65.61.

= 65 − 64 = 1, 65.61 − 65 = 0.61

0.61 < 1: closer to 8.1

So √65 is about 8.1, to the nearest tenth.

= √65 ≈ 8.1

6 Check

Let's check with the number halfway: 8.05.

8.05 times 8.05 is 64.8025, less than 65.

8.05 × 8.05 = 64.8025 < 65

So √65 is above 8.05.

To the nearest tenth, it is 8.1.

The answer is right!

√65 > 8.05: nearest tenth 8.1

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