Step by step
The question
Question: estimate √40.
The square root of 40 is the number that times itself makes 40.
Careful!
The square root is not half.
20 times 20 is 400, far more than 40.
1 List the squares
First, the square numbers near 40.
5 times 5 is 25.
6 times 6 is 36.
7 times 7 is 49.
8 times 8 is 64.
2 Find where it fits
40 is between the squares 36 and 49.
A bigger number has a bigger square root.
So √40 is between √36 and √49.
√36 is 6.
√49 is 7.
So √40 is between 6 and 7.
3 Which is closer
40 is 4 more than 36, and 9 less than 49.
4 is less.
So 40 is closer to 36, and √40 is closer to 6.
So √40 is between 6 and 7, closer to 6.
4 Check
Let's check with the number halfway: 6.5.
6.5 times 6.5 is 42.25, more than 40.
So √40 is below 6.5: closer to 6.
The answer is right!
Let's look back.
First: list the squares.
Then: find where it fits.
Then: which is closer.
Last: check.