Step by step
The question
Question: simplify √50 + √18.
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 50.
A number that goes in, other than 50 itself, is at most half of it.
So we list them up to half of 50.
First 50.
We try the biggest first.
Does 25 go into 50?
Yes: 50 divided by 25 is 2.
Now 18.
Does 9 go into 18?
Yes: 18 divided by 9 is 2.
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 √50 is √25 times √2, and √18 is √9 times √2.
√25 is 5, because 5 times 5 is 25.
√9 is 3, because 3 times 3 is 9.
We write 5 in place of √25, and 3 in place of √9.
We leave out the times sign: 5√2 means 5 times √2.
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 √2.
They are like terms, like 5 and 3 of the same thing.
5 plus 3 is 8.
So √50 + √18 = 8√2.
4 Check
Let's check with a calculator.
The start is about 11.31.
The answer is about 11.31 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.