Step by step
The question
Question: simplify √5 over √8.
1 Simplify the bottom first
Simplify the bottom root first.
It keeps the numbers small.
8 is 4 times 2, and √4 is 2.
So √8 is 2√2.
So the bottom is 2√2.
2 Remove the root at the bottom
A root times itself is the number inside: √2 times √2 is 2.
So we multiply the bottom by √2.
To keep the fraction the same, we multiply the top by √2 too.
√2 over √2 is 1, and times 1 changes nothing.
So we multiply by √2 over √2.
Careful!
Many people multiply only the bottom.
Then the fraction changes.
Top and bottom get the same √2.
On top: √5 times √2 is √10.
On the bottom: 2 times 2 is 4.
No root at the bottom now.
The numbers can't be cancelled, so we are done.
3 Check
Let's check with a calculator.
The start is about 0.79.
The answer is about 0.79 too.
So it is right!
Let's look back.
First: simplify the bottom first.
Then: remove the root at the bottom.
Last: check.