Step by step
The question
Question: simplify √3 over √5.
√3√5
1 Remove the root at the bottom
A root times itself is the number inside: √5 times √5 is 5.
So we multiply the bottom by √5.
√5 × √5 = 5
To keep the fraction the same, we multiply the top by √5 too.
√5 over √5 is 1, and times 1 changes nothing.
√5√5 = 1
So we multiply by √5 over √5.
= √3√5 × √5√5
Careful!
Many people multiply only the bottom.
Then the fraction changes.
Top and bottom get the same √5.
Careful: multiply the TOP by the root too
On top: √3 times √5 is √15.
On the bottom: √5 times √5 is 5.
No root at the bottom now.
The numbers can't be cancelled, so we are done.
= √155
2 Check
Let's check with a calculator.
The start is about 0.77.
Start: √3√5 ≈ 0.77
The answer is about 0.77 too.
So it is right!
Answer: √155 ≈ 0.77
Let's look back.
First: remove the root at the bottom.
Last: check.
Answer
= √155