Step by step
The question
Question: simplify √(27y³).
√27y3
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 27 is 9: 27 is 9 times 3.
A square of letters has an even power.
y³ is y² times y.
y² is y squared, because y times y is y²: the powers add.
27 = 9 × 3
y3 = y2 × y, y² = (y)²
We write the squares first, then what is left.
= √9y2 × 3y
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.
= √9y2 × √3y
√9 is 3.
The root halves each even power: √y² is y.
√9 = 3, √y² = y
So √(27y³) = 3y√3y.
= 3y√3y
3 Check
Let's check: we square the answer.
We must get 27y³ back.
A root times itself is what is inside.
Squaring doubles each power.
Check: (3y√3y)2 = (3y)2 × 3y = 9y2 × 3y
= 27y3
We get 27y³.
So 3y√3y is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.
Answer
= 3y√3y