Step by step
The question
Question: which expression is equivalent to √72?
Equivalent means: the same value for every x.
Simplify means: take every square number out from under the root.
There are four choices.
First we simplify it, the normal way.
Then we pick.
1 Find the biggest square number
Square numbers are a number times itself: 2 × 2 = 4, 3 × 3 = 9, and so on.
We want one that goes into 72.
A number that goes in, other than 72 itself, is at most half of it.
So we list them up to half of 72.
We try the biggest first.
Does 36 go into 72?
Yes: 72 divided by 36 is 2.
2 Split the root
So 72 is 36 times 2.
We write it under the root.
Rule: when two numbers are multiplied under a root, we can split it into two roots.
So √(36 × 2) is √36 times √2.
√36 is 6, because 6 times 6 is 36.
We write 6 in place of √36.
We leave out the times sign: 6√2 means 6 times √2.
2 has no square number in it, so we are done.
So √72 = 6√2.
3 Check
Let's check.
√72 times √72 is 72.
So the answer times itself must give 72 too.
Squared means times itself.
6 × 6 is 36, and √2 × √2 is 2.
36 × 2 is 72.
We get 72.
So 6√2 is right!
4 Pick the choice
Our answer is 6√2.
That is choice B.
5 Why not the others
A is 2√6.
Wrong: 72 = 36 × 2, and it is the 36 that comes out, as 6.
C is 6√12.
Wrong: 72 = 6 × 12: took the 6 out, but 6 is not a square number.
D is 36√2.
Wrong: the 36 came out as 36: √36 is 6.
Let's look back.
First: find the biggest square number.
Then: split the root.
Then: check.
Then: pick the choice.
Last: why not the others.