Cube Root of Negative Numbers — Examples & Practice

A negative number's cube root will always be negative

When working with exponents, the only way to get negative results is to raise a negative number to an odd power. Since cubing a number means raising it to the $3^{\text{rd}}$ power—which is odd—the cube roots of negative numbers must also be negative.

This applies to perfect cubes and estimating other cube roots. Let's look at a perfect cube example because it's easier to write out. We are going to find $\sqrt[3]{-64}.$

$$-4 = (-4)^1 \text{ is negative}$$ $$-4 \times -4 = (-4)^2 = 16 \text{ is positive}$$ $$-4 \times -4 \times -4= (-4)^3 = -64 \text{ is negative}$$

When the exponent is odd (like $1$ or $3$), the result is negative. When exponent is even (like $2$), the result is positive. Cubes always involve multiplying a number by itself three times, so if the number is negative the cube will always be negative (and if the cube is negative, its cube root has to be negative too).

The cube root of a negative number is really easy if you remember one thing.

And this is the key to taking the cube root of negative integers. For example, if I wanted to take the cube root of $-125$, I would see that this number is really $-1\cdot125$ so its cube root is just $\sqrt[3]{-1}\cdot\sqrt[3]{125}=-1\cdot5=-5$. The cube root of a negative number is negative!

In fact, this method actually works for any odd root. As long as you remember that the fifth, seventh, etc. root of a negative number is also negative, you should now be able to take almost any root.

(Video) Learning how to take the cube root of a negative number, cube root(-27)

by Brian McLogan

This video by Brian McLogan covers how to take the cubed root of a negative number. For example, $\sqrt[3]{-27}$.

$\large{\text{Summary}}$

Negative numbers can't be square roots, but they can be cubed roots because multiplying an odd number of negative numbers gives a negative number.

What number multiplied by itself $3$ times is $-27$?

It is either $-3\times-3\times-3$ or $3\times3\times3$. It has to be $-3\times-3\times-3$.