Step by step
The question
Question: simplify √(18x³).
√18x3
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 18 is 9: 18 is 9 times 2.
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.
18 = 9 × 2
x3 = x2 × x, x² = (x)²
We write the squares first, then what is left.
= √9x2 × 2x
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.
= √9x2 × √2x
√9 is 3.
The root halves each even power: √x² is x.
√9 = 3, √x² = x
So √(18x³) = 3x√2x.
= 3x√2x
3 Check
Let's check: we square the answer.
We must get 18x³ back.
A root times itself is what is inside.
Squaring doubles each power.
Check: (3x√2x)2 = (3x)2 × 2x = 9x2 × 2x
= 18x3
We get 18x³.
So 3x√2x is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.
Answer
= 3x√2x