Step by step
The question
Question: simplify √18 + √32 − √2.
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 32.
A bigger one can't go in.
First 18.
We try the biggest first.
Does 9 go into 18?
Yes: 18 divided by 9 is 2.
Now 32.
Does 16 go into 32?
Yes: 32 divided by 16 is 2.
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.
√9 is 3, because 3 times 3 is 9.
√16 is 4, because 4 times 4 is 16.
We write the numbers in place of the square roots.
We leave out the times sign: 3√2 means 3 times √2.
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 √2.
They are like terms, like 3 and 4 and 1 of the same thing.
3 plus 4 minus 1 is 6.
So √18 + √32 − √2 = 6√2.
4 Check
Let's check with a calculator.
The start is about 8.49.
The answer is about 8.49 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.