Step by step
The question
Question: simplify √(8x⁷).
Like the books, we take every letter as a number that is not negative.
1 Find the squares
The biggest square in 8 is 4: 8 is 4 times 2.
A square of letters has an even power.
x⁷ is x⁶ times x.
x⁶ is x³ squared, because x³ times x³ is x⁶: 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.
√4 is 2.
The root halves each even power: √x⁶ is x³.
Careful!
The root does not take 2 off the power.
It halves it: x³ times x³ is x⁶.
So √(8x⁷) = 2x³√2x.
3 Check
Let's check: we square the answer.
We must get 8x⁷ back.
A root times itself is what is inside.
Squaring doubles each power.
We get 8x⁷.
So 2x³√2x is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.