Step by step
The question
Question: estimate √99 to the nearest tenth.
The square root of 99 is the number that times itself makes 99.
Careful!
The square root is not half.
49.5 times 49.5 is 2450.25, far more than 99.
1 List the squares
First, the square numbers near 99.
8 times 8 is 64.
9 times 9 is 81.
10 times 10 is 100.
11 times 11 is 121.
2 Find where it fits
99 is between the squares 81 and 100.
A bigger number has a bigger square root.
So √99 is between √81 and √100.
√81 is 9.
√100 is 10.
So √99 is between 9 and 10.
3 Which is closer
99 is 18 more than 81, and 1 less than 100.
1 is less.
So 99 is closer to 100, and √99 is closer to 10.
4 Try tenths
√99 is a bit less than 10.
So we try 9.9 and so on, and square each one.
9.9 times 9.9 is 98.01.
That is less than 99: too small.
So √99 is between 9.9 and 10.0.
5 Pick the nearest tenth
99 is 0.99 more than 98.01, and 1 less than 100.
So it is closer to 98.01.
So √99 is about 9.9, to the nearest tenth.
6 Check
Let's check with the number halfway: 9.95.
9.95 times 9.95 is 99.0025, more than 99.
So √99 is below 9.95.
To the nearest tenth, it is 9.9.
The answer is right!
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.