Step by step
The question
Question: simplify √75 − √12.
Simplify means: take every square number out from under the root.
1 Find a square number in each
Square numbers are a number times itself: 2 × 2 = 4, 3 × 3 = 9, and so on.
We want one that goes into 75.
A number that goes in, other than 75 itself, is at most half of it.
So we list them up to half of 75.
First 75.
We try the biggest first.
Does 36 go into 75?
No.
Does 25 go into 75?
Yes: 75 divided by 25 is 3.
Now 12.
Does 4 go into 12?
Yes: 12 divided by 4 is 3.
2 Split each root
We write each number as the square times what is left.
Rule: when two numbers are multiplied under a root, we can split it into two roots.
So √75 is √25 times √3, and √12 is √4 times √3.
√25 is 5, because 5 times 5 is 25.
√4 is 2, because 2 times 2 is 4.
We write 5 in place of √25, and 2 in place of √4.
We leave out the times sign: 5√3 means 5 times √3.
Careful!
We can't add the numbers under the roots.
Only roots of the same number go together.
3 Put like roots together
They all have √3.
They are like terms, like 5 and 2 of the same thing.
5 minus 2 is 3.
So √75 − √12 = 3√3.
4 Check
Let's check with a calculator.
The start is about 5.20.
The answer is about 5.20 too.
So it is right!
Let's look back.
First: find a square number in each.
Then: split each root.
Then: put like roots together.
Last: check.