Step by step
The question
Question: simplify √(x³).
√x3
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.
x³ is x² times x.
x² is x squared, because x times x is x²: the powers add.
x3 = x2 × x, x² = (x)²
We write the squares first, then what is left.
= √x2 × x
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.
= √x2 × √x
The root halves each even power: √x² is x.
√x² = x
So √(x³) = x√x.
= x√x
3 Check
Let's check: we square the answer.
We must get x³ back.
A root times itself is what is inside.
Squaring doubles each power.
Check: (x√x)2 = (x)2 × x = x2 × x = x3
We get x³.
So x√x is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.
Answer
= x√x