Step by step
The question
Question: simplify √(x⁴y²).
√x4y2
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.
y² is y squared, because y times y is y²: the powers add.
x4 = x4, x⁴ = (x2)²
y2 = y2, y² = (y)²
Everything under the root is a square.
= √x4y2
2 Take the square roots
The root halves each even power: √x⁴ is x², √y² is y.
√x⁴ = x², √y² = y
Careful!
The root does not take 2 off the power.
It halves it: x² times x² is x⁴, y times y is y².
Careful: not √x⁴ = x², √y² = y⁰ ✗
So √(x⁴y²) = x²y.
= x2y
3 Check
Let's check: we square the answer.
We must get x⁴y² back.
Squaring doubles each power.
Check: (x2y)2 = x4y2
We get x⁴y².
So x²y is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.
Answer
= x2y