Graph of $x^{(2/3)}$

Question

While examining the basics of basics of math, thinking about it non-conventionally, I stumbled upon the following dilemma:

When given a function: $$f(x) = \left(\sqrt[3]{x}\right)^2$$

Why does Wolfram draw this?

Why is the real part of the function negative for $x<0$?

Isn't $f(-8)$ for example a legit, real value of 4?

Answer 1

WolframAlpha is graphing $x^{2/3}$ using complex values. One value for $(-8)^{2/3}$ is $4$, but another value is $-2+2i\sqrt{3}$.

If you read your graph, you will see the complex value is displayed. Read the $x$-axis as your input (on the $x$-axis, go to $-8$) and read the real and imaginary parts as the output (real part is $-2$, imaginary part is about $3.5$).

Change the plot from complex-valued to real-valued, and change from principal root to real-valued root, and you'll generate the graph that you expect.