What Is the Cube Root of a Negative Number?

A negative number has a perfectly well-defined real cube root. For a positive number $a$, the real cube root satisfies

$$\sqrt[3]{-a}=-\sqrt[3]{a}.$$

For example, $\sqrt[3]{-27}=-3$ because $(-3)^3=-27$. This differs from square roots: no real number has a square equal to a negative number, but negative numbers do have real cube roots because cubing preserves a number’s sign.

The definition of a cube root

The cube root of $x$ is a number $y$ such that

$$y^3=x.$$

For real numbers, every $x$ has exactly one real cube root. The function $y\mapsto y^3$ is continuous and strictly increasing on the real line, with values tending to $-\infty$ as $y\to-\infty$ and to $+\infty$ as $y\to+\infty$. Therefore, it takes each real value once and only once.

If $x>0$, its cube root is positive. If $x=0$, its cube root is $0$. If $x<0$, its cube root is negative. Thus the sign rule is

$$\sqrt[3]{x} = \begin{cases} \text{a positive number}, & x>0,\\ 0, & x=0,\\ \text{a negative number}, & x<0. \end{cases}$$

Why the cube root of a negative number is negative

Let $a>0$. If $r=\sqrt[3]{a}$, then $r^3=a$. Negating $r$ gives

$$(-r)^3=(-r)(-r)(-r)=-r^3=-a.$$

Therefore $-r$ is the real number whose cube is $-a$, which proves

$$\sqrt[3]{-a}=-\sqrt[3]{a}.$$

This is a special property of odd roots. More generally, for an odd positive integer $n$,

$$\sqrt[n]{-a}=-\sqrt[n]{a} \qquad (a>0).$$

Even roots behave differently. For example, $\sqrt{-9}$ is not a real number because no real square is negative.

Worked examples

Example 1: An exact cube

Evaluate $\sqrt[3]{-125}$.

Since $125=5^3$,

$$\sqrt[3]{-125}=-\sqrt[3]{125}=-5.$$

Check the result:

$$(-5)^3=(-5)(-5)(-5)=-125.$$

Example 2: A number that is not a perfect cube

To estimate $\sqrt[3]{-20}$, first observe that

$$2^3=8, \qquad 3^3=27.$$

So $\sqrt[3]{20}$ lies between $2$ and $3$, and the negative cube root lies between $-3$ and $-2$:

$$-3<\sqrt[3]{-20}<-2.$$

A calculator gives approximately

$$\sqrt[3]{-20}\approx -2.7144.$$

Cubing this approximation gives a value close to $-20$; the small discrepancy comes from rounding.

Example 3: Solving an equation

Solve

$$x^3=-64$$

over the real numbers. Taking the real cube root of both sides gives

$$x=\sqrt[3]{-64}=-4.$$

Indeed, $(-4)^3=-64$.

Real cube root versus complex cube roots

The answer depends on whether the problem is asking for a real root or all roots over the complex numbers.

For the equation

$$z^3=-a, \qquad a>0,$$

there are three complex solutions. Let $r=\sqrt[3]{a}$. The three solutions are

$$-r, \qquad \frac{r}{2}+\frac{\sqrt{3}}{2}ri, \qquad \frac{r}{2}-\frac{\sqrt{3}}{2}ri.$$

The first solution, $-r$, is the real cube root. The other two are complex conjugates. For example, the solutions of $z^3=-8$ are

$$-2, \qquad 1+\sqrt{3}i, \qquad 1-\sqrt{3}i.$$

You can verify the nonreal roots using the polar form of complex numbers. Since

$$-8=8e^{i(\pi+2\pi k)},$$

the cube roots are

$$2e^{i(\pi+2\pi k)/3}, \qquad k=0,1,2.$$

These produce the three values above.

In elementary real-number problems, the notation $\sqrt[3]{-8}$ normally means the unique real value $-2$. In complex analysis, a phrase such as the principal cube root may refer to a particular complex branch. For a negative real input, that principal complex value is generally nonreal, so the convention should be stated when it matters.

Common mistakes

Dropping the negative sign

It is incorrect to write

$$\sqrt[3]{-27}=3.$$

The number $3$ cubes to $27$, not $-27$. The correct real value is $-3$.

Confusing cube roots with square roots

The fact that $\sqrt{-1}$ is not real does not imply that $\sqrt[3]{-1}$ is not real. In fact,

$$\sqrt[3]{-1}=-1.$$

Applying the wrong exponent rule

For real numbers, the notation $x^{1/3}$ agrees with the real cube root when interpreted using the real-valued odd root:

$$(-8)^{1/3}=-2.$$

However, fractional powers in complex arithmetic can depend on a chosen logarithm branch. Do not automatically transfer real exponent rules to complex powers without checking the convention.

Forgetting the question’s number system

If a problem asks for the real cube root, give the single real answer. If it asks for all complex solutions of a cubic equation, include all three roots, including the two nonreal ones.