I want to get a figure for $(-x)^{\frac23}$. After using the following code, I found that only the left part of the function $(-x)^{\frac23}$ was plotted.
Plot[(-x)^(2/3), {x, -30, 30}]
Here is a image of the output:
How can I get the right plot?
As the cube root of a number is potentially multivalued (depending on the context you are using it in) Mathematica uses the following conventions to differentiate between the different roots of $-1$:
$-1$ represents the real number $-1$. I.e. the root with argument of $0$.
$(-1)^\frac13$ represents the complex number $\frac12+\frac{\sqrt{3}}{2}i$. I.e. the root with argument of $\frac{\pi}{3}$.
$(-1)^\frac23$ represents the complex number $-\frac12+\frac{\sqrt{3}}{2}i$. I.e. the root with argument of $\frac{2\pi}{3}$.
So when you ask it to evaluate $(-x)$^$(2/3)$ you have inadvertently using the last notation and Mathematica thinks you want the complex answer with argument of $\frac{2\pi}{3}$. Hence why nothing is plotted.
If you wanted Mathematica to graph the real value answer you can get around it with either of the following commands:
This will make it evaluate the squaring first before doing the cube root. Squaring first ensures the value being cube rooted is non-negative so Mathematica will only return a single value (the real root).
When applied to complex number the Abs function will find the modulus of the number (like how we use $|x|$ on the domains of reals vs complex).