Step by step
The question
Question: simplify √12 + √27.
1 Find a square number in each
Square numbers are a number times itself: 2 × 2 = 4, 3 × 3 = 9, and so on.
We list them up to half of 27.
A bigger one can't go in.
First 12.
We try the biggest first.
Does 4 go into 12?
Yes: 12 divided by 4 is 3.
Now 27.
Does 9 go into 27?
Yes: 27 divided by 9 is 3.
2 Split each root
We write each number as the square times what is left.
Rule: a root over a times can be split into two roots.
So each root becomes two roots, times.
√4 is 2, because 2 times 2 is 4.
√9 is 3, because 3 times 3 is 9.
We write the numbers in place of the square roots.
We leave out the times sign: 2√3 means 2 times √3.
Careful!
We can't add the numbers under the roots.
√9 + √16 is 3 + 4 = 7, not √25 = 5.
3 Put like roots together
They all have √3.
They are like terms, like 2 and 3 of the same thing.
2 plus 3 is 5.
So √12 + √27 = 5√3.
4 Check
Let's check with a calculator.
The start is about 8.66.
The answer is about 8.66 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.