Step by step
The question
Question: simplify √(50x⁴y³).
Like the books, we take every letter as a number that is not negative.
1 Find the squares
The biggest square in 50 is 25: 50 is 25 times 2.
A square of letters has an even power.
x⁴ is x² squared, because x² times x² is x⁴: the powers add.
y³ is y² times y.
y² is y squared, because y times y is y²: 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.
√25 is 5.
The root halves each even power: √x⁴ is x², √y² is y.
Careful!
The root does not take 2 off the power.
It halves it: x² times x² is x⁴, y times y is y².
So √(50x⁴y³) = 5x²y√2y.
3 Check
Let's check: we square the answer.
We must get 50x⁴y³ back.
A root times itself is what is inside.
Squaring doubles each power.
We get 50x⁴y³.
So 5x²y√2y is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.