What is the value of $(-1)^\frac{4}{3}$?

Question

I was trying to plot the graph of $y=x^\frac{4}{3}$. However, I tried two online plotters, both gave me curves only on the right side of the y-axis. There is nothing on the left side of the y-axis. Shouldn't it be a curve symmetric to the y-axis? Similarily for $y=x^\frac{5}{3}$ which I thought is a function symmetric to origin but only has value for nonnegative x.

I tried to use google to compute $(-1)^\frac{4}{3}$ and it automatically gives me $-0.5 - 0.866025404 i$ instead of $1$. And $(-1)^\frac{5}{3}$ got a answer of $0.5 - 0.866025404 i$ instead of $-1$. Why does the result include an imaginary part?

Answer 1

Fractional powers of negative numbers aren't uniquely defined.

There are three cube roots of $-1$: $-1$, $\frac12+\frac{\sqrt{3}}2i$, and $\frac12-\frac{\sqrt3}2i$.

The answer given by Google for $(-1)^{4/3}$ was the fourth power of the middle one of those three.

Answer 2

The function is defined for positive and negative values of $x$. The graph shows the real and imaginary parts:

$${\rm Re}[(-1)^{4/3}] = -\frac{1}{2}$$

$${\rm Im}[(-1)^{4/3}] = - \frac{\sqrt{3}}{2}$$

Given that there is no unique way to compute a partial root of a negative number, Mathematica seems to assume the most general complex form.

There is no reason that the function should be (or is) symmetric with respect to the interchange $x \leftrightarrow -x$.

Answer 3

$$\displaystyle (-1)^{4/3}=\left[e^{(2k-1)\pi i}\right]^{4/3}, \quad k\in \{0,1,2\}$$

$$\displaystyle (-1)^{4/3} \in \left\{ e^{-\frac{4\pi i}{3}}, e^\frac{4\pi i}{3}, e^{4\pi i} \right\}$$

We can write this in rectangular coordinates:

$$\displaystyle (-1)^{4/3} \in \left\{- \frac{1}{2}+\frac{\sqrt{3}}{2}, -\frac{1}{2}-\frac{\sqrt{3}}{2}, 1 \right\}$$