Worked examples
The question
Question: simplify √(x⁴).
√x4
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.
x⁴ is x² squared, because x² times x² is x⁴: the powers add.
x4 = x4, x⁴ = (x2)²
Everything under the root is a square.
= √x4
2 Take the square roots
The root halves each even power: √x⁴ is x².
√x⁴ = x²
Careful!
The root does not take 2 off the power.
It halves it: x² times x² is x⁴.
Careful: not √x⁴ = x² ✗
So √(x⁴) = x².
= x2
3 Check
Let's check: we square the answer.
We must get x⁴ back.
Squaring doubles each power.
Check: (x2)2 = x4
We get x⁴.
So x² is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.
▶ Watch it solved
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.
▶ Watch it solved
The question
Question: simplify √(50x⁴y³).
√50x4y3
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 50 is 25: 50 is 25 times 2.
A square of letters has an even power.
x⁴ is x² squared, because x² times x² is x⁴: the powers add.
y³ is y² times y.
y² is y squared, because y times y is y²: the powers add.
50 = 25 × 2
x4 = x4, x⁴ = (x2)²
y3 = y2 × y, y² = (y)²
We write the squares first, then what is left.
= √25x4y2 × 2y
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.
= √25x4y2 × √2y
√25 is 5.
The root halves each even power: √x⁴ is x², √y² is y.
√25 = 5, √x⁴ = x², √y² = y
Careful!
The root does not take 2 off the power.
It halves it: x² times x² is x⁴, y times y is y².
Careful: not √x⁴ = x², √y² = y⁰ ✗
So √(50x⁴y³) = 5x²y√2y.
= 5x2y√2y
3 Check
Let's check: we square the answer.
We must get 50x⁴y³ back.
A root times itself is what is inside.
Squaring doubles each power.
Check: (5x2y√2y)2 = (5x2y)2 × 2y
= 25x4y2 × 2y = 50x4y3
We get 50x⁴y³.
So 5x²y√2y is right!
Let's look back.
First: find the squares.
Then: take the square roots.
Last: check.
▶ Watch it solved
