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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?

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    Try [`cbrt(x)^2`](http://www.wolframalpha.com/input/?i=cbrt(x)%5E2) instead. WA interpretes $(\sqrt[3]{x})^2$ as the principal (complex) value. – 2017-02-12
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    You are correct: WolframAlpha is skittish when it comes to fractional powers of negative numbers. To maintain full generality, it tends to use the Principal Root, which is a complex-valued function defined over complex numbers. But this generality is not appropriate when the domain is $\mathbb R$ and the exponent is a rational number with an odd denominator. – 2017-02-12
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    And I've always used such programs as the ultimate proof of calculations... Apparently, you can trust nothing at all. – 2017-02-12
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    @DartDega Trust, but verify ;-) If you enter [`x^(2/3)`](http://www.wolframalpha.com/input/?i=x%5E(2%2F3)) in WA, the very next line displays "***Assuming the principal root $\mid$ Use the real‐valued root instead***". – 2017-02-12
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    @dxiv To be honest - I didn't notice the switch. After switching it though, an interesting behavior remains - this time the domain is being narrowed down to positive numbers. Why is it so? – 2017-02-12
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    @DartDega I don't see that [here](http://www.wolframalpha.com/input/?i=x%5E(2%2F3)&rawformassumption=%22%5E%22+-%3E+%22Real%22). Anyway, you can always specify the domain explicitly e.g. [`x^(2/3) for x=-10 to 10`](http://www.wolframalpha.com/input/?i=x%5E(2%2F3)+for+x%3D-10+to+10&rawformassumption=%22%5E%22+-%3E+%22Real%22). – 2017-02-12
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    @dxiv Interesting, only the mobile version of Wolfram shows a half of the graph (http://imgur.com/HrHb9ad). Anyway, I guess my question is answered. Thank you very much. – 2017-02-12
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    @DartDega Interesting, indeed, guess I learned something new as well. – 2017-02-12

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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.