Step by step
The question
Question: simplify √(12x⁵).
Like the books, we take every letter as a number that is not negative.
1 Find the squares
The biggest square in 12 is 4: 12 is 4 times 3.
A square of letters has an even power.
x⁵ is x⁴ times x.
x⁴ is x² squared, because x² times x² is x⁴: 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.
√4 is 2.
The root halves each even power: √x⁴ is x².
Careful!
The root does not take 2 off the power.
It halves it: x² times x² is x⁴.
So √(12x⁵) = 2x²√3x.
3 Check
Let's check: we square the answer.
We must get 12x⁵ back.
A root times itself is what is inside.
Squaring doubles each power.
We get 12x⁵.
So 2x²√3x is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.