Step by step
The question
Question: simplify √(x⁸).
√x8
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.
x⁸ is x⁴ squared, because x⁴ times x⁴ is x⁸: the powers add.
x8 = x8, x⁸ = (x4)²
Everything under the root is a square.
= √x8
2 Take the square roots
The root halves each even power: √x⁸ is x⁴.
√x⁸ = x⁴
Careful!
The root does not take 2 off the power.
It halves it: x⁴ times x⁴ is x⁸.
Careful: not √x⁸ = x⁶ ✗
So √(x⁸) = x⁴.
= x4
3 Check
Let's check: we square the answer.
We must get x⁸ back.
Squaring doubles each power.
Check: (x4)2 = x8
We get x⁸.
So x⁴ is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.
Answer
= x4