Step by step
The question
Question: simplify √(y⁵).
√y5
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.
y⁵ is y⁴ times y.
y⁴ is y² squared, because y² times y² is y⁴: the powers add.
y5 = y4 × y, y⁴ = (y2)²
We write the squares first, then what is left.
= √y4 × y
2 Take the square roots
Rule: a root over a times can be split into two roots.
√(a × b) = √a × √b
So we get two roots.
= √y4 × √y
The root halves each even power: √y⁴ is y².
√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 √(y⁵) = y²√y.
= y2√y
3 Check
Let's check: we square the answer.
We must get y⁵ back.
A root times itself is what is inside.
Squaring doubles each power.
Check: (y2√y)2 = (y2)2 × y = y4 × y = y5
We get y⁵.
So y²√y is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.
Answer
= y2√y