Step by step
The question
Question: simplify √(20a⁶b).
Like the books, we take every letter as a number that is not negative.
1 Find the squares
The biggest square in 20 is 4: 20 is 4 times 5.
A square of letters has an even power.
a⁶ is a³ squared, because a³ times a³ is a⁶: the powers add.
b on its own has no square in it.
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.
√4 is 2.
The root halves each even power: √a⁶ is a³.
Careful!
The root does not take 2 off the power.
It halves it: a³ times a³ is a⁶.
So √(20a⁶b) = 2a³√5b.
3 Check
Let's check: we square the answer.
We must get 20a⁶b back.
A root times itself is what is inside.
Squaring doubles each power.
We get 20a⁶b.
So 2a³√5b is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.