Step by step
The question
Question: simplify √(4x⁶).
√4x6
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 4 is 4: 4 is 4 times 1.
A square of letters has an even power.
x⁶ is x³ squared, because x³ times x³ is x⁶: the powers add.
4 = 4 × 1
x6 = x6, x⁶ = (x3)²
Everything under the root is a square.
= √4x6
2 Take the square roots
√4 is 2.
The root halves each even power: √x⁶ is x³.
√4 = 2, √x⁶ = x³
Careful!
The root does not take 2 off the power.
It halves it: x³ times x³ is x⁶.
Careful: not √x⁶ = x⁴ ✗
So √(4x⁶) = 2x³.
= 2x3
3 Check
Let's check: we square the answer.
We must get 4x⁶ back.
Squaring doubles each power.
Check: (2x3)2 = 4x6
We get 4x⁶.
So 2x³ is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.
Answer
= 2x3