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