Step by step
The question
Question: estimate √200.
The square root of 200 is the number that times itself makes 200.
Careful!
The square root is not half.
100 times 100 is 10000, far more than 200.
1 List the squares
First, the square numbers near 200.
13 times 13 is 169.
14 times 14 is 196.
15 times 15 is 225.
16 times 16 is 256.
2 Find where it fits
200 is between the squares 196 and 225.
A bigger number has a bigger square root.
So √200 is between √196 and √225.
√196 is 14.
√225 is 15.
So √200 is between 14 and 15.
3 Which is closer
200 is 4 more than 196, and 25 less than 225.
4 is less.
So 200 is closer to 196, and √200 is closer to 14.
So √200 is between 14 and 15, closer to 14.
4 Check
Let's check with the number halfway: 14.5.
14.5 times 14.5 is 210.25, more than 200.
So √200 is below 14.5: closer to 14.
The answer is right!
Let's look back.
First: list the squares.
Then: find where it fits.
Then: which is closer.
Last: check.