Step by step
The question
Question: simplify √(45x⁹).
Like the books, we take every letter as a number that is not negative.
1 Find the squares
The biggest square in 45 is 9: 45 is 9 times 5.
A square of letters has an even power.
x⁹ is x⁸ times x.
x⁸ is x⁴ squared, because x⁴ times x⁴ is x⁸: 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.
√9 is 3.
The root halves each even power: √x⁸ is x⁴.
Careful!
The root does not take 2 off the power.
It halves it: x⁴ times x⁴ is x⁸.
So √(45x⁹) = 3x⁴√5x.
3 Check
Let's check: we square the answer.
We must get 45x⁹ back.
A root times itself is what is inside.
Squaring doubles each power.
We get 45x⁹.
So 3x⁴√5x is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.