Step by step
The question
Question: simplify 3√20.
1 Find the biggest square number
Square numbers are a number times itself: 2 × 2 = 4, 3 × 3 = 9, and so on.
We list them up to half of 20.
A bigger one can't go in.
We try the biggest first.
Does 9 go into 20?
No.
Does 4 go into 20?
Yes: 20 divided by 4 is 5.
2 Split the root
So 20 is 4 times 5.
We write it under the root.
3√20 means 3 times √20.
Rule: a root over a times can be split into two roots.
So √(4 × 5) is √4 times √5.
√4 is 2, because 2 times 2 is 4.
We write 2 in place of √4.
3 times 2 is 6.
We leave out the times sign: 6√5 means 6 times √5.
5 has no square number in it, so we are done.
So 3√20 = 6√5.
3 Check
Let's check: we square both.
They must give the same number.
Squared means times itself.
First the start: 3 × 3, and √20 × √20 = 20.
Now the answer: 6 × 6, and √5 × √5 = 5.
Both give 180.
So 6√5 is right!
Let's look back.
First: find the biggest square number.
Then: split the root.
Last: check.