Step by step
The question
Question: simplify √(x⁶).
√x6
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.
x⁶ is x³ squared, because x³ times x³ is x⁶: the powers add.
x6 = x6, x⁶ = (x3)²
Everything under the root is a square.
= √x6
2 Take the square roots
The root halves each even power: √x⁶ is x³.
√x⁶ = x³
Careful!
The root does not take 2 off the power.
It halves it: x³ times x³ is x⁶.
Careful: not √x⁶ = x⁴ ✗
So √(x⁶) = x³.
= x3
3 Check
Let's check: we square the answer.
We must get x⁶ back.
Squaring doubles each power.
Check: (x3)2 = x6
We get x⁶.
So x³ is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.
Answer
= x3