Step by step
The question
Question: simplify √(25y⁴).
√25y4
Like the books, we take every letter as a number that is not negative.
letters are not negative here
1 Find the squares
The biggest square in 25 is 25: 25 is 25 times 1.
A square of letters has an even power.
y⁴ is y² squared, because y² times y² is y⁴: the powers add.
25 = 25 × 1
y4 = y4, y⁴ = (y2)²
Everything under the root is a square.
= √25y4
2 Take the square roots
√25 is 5.
The root halves each even power: √y⁴ is y².
√25 = 5, √y⁴ = y²
Careful!
The root does not take 2 off the power.
It halves it: y² times y² is y⁴.
Careful: not √y⁴ = y² ✗
So √(25y⁴) = 5y².
= 5y2
3 Check
Let's check: we square the answer.
We must get 25y⁴ back.
Squaring doubles each power.
Check: (5y2)2 = 25y4
We get 25y⁴.
So 5y² is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.
Answer
= 5y2