Step by step
The question
Question: solve x³ = −64.
x3 = −64
x³ means x times x times x.
We want just x.
x³ = x × x × x
1 Cube root of both sides
The cube root asks: which number, times itself three times, makes this?
∛27 = ? → ? × ? × ? = 27
The cubes: 1 times 1 times 1 is 1.
2 times 2 times 2 is 8.
3 times 3 times 3 is 27.
4 times 4 times 4 is 64.
1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64
x is the cube root of −64.
x = 3√−64
A negative number, times itself three times, stays negative: minus times minus is plus, times minus again is minus.
(−4)³ = − × − × − = −
−4 times −4 times −4 is −64.
So x = −4.
x = −4
Careful!
With squares, a negative has no root.
With cubes it does: the answer is negative.
Careful: a negative number has a cube root
2 Check
Let's check: −4 times −4 times −4 is −64.
(−4) × (−4) × (−4) = −64
The same as the right side.
So the answer is right!
Let's look back.
First: cube root of both sides.
Last: check.
Answer
Careful: a negative number has a cube root