Step by step
The question
Question: simplify √(72x²y⁵).
Like the books, we take every letter as a number that is not negative.
1 Find the squares
The biggest square in 72 is 36: 72 is 36 times 2.
A square of letters has an even power.
x² is x squared, because x times x is x²: the powers add.
y⁵ is y⁴ times y.
y⁴ is y² squared, because y² times y² is y⁴: the powers add.
We write the squares first, then what is left.
2 Take the square roots
Rule: a root over a times can be split into two roots.
So we get two roots.
√36 is 6.
The root halves each even power: √x² is x, √y⁴ is y².
Careful!
The root does not take 2 off the power.
It halves it: x times x is x², y² times y² is y⁴.
So √(72x²y⁵) = 6xy²√2y.
3 Check
Let's check: we square the answer.
We must get 72x²y⁵ back.
A root times itself is what is inside.
Squaring doubles each power.
We get 72x²y⁵.
So 6xy²√2y is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.