Consider the following: $$x^{2/3}$$
I am having some really trivial doubts, but I would like to clarify them all, once for all. My questions are:
Is $x^{2/3}$ the same as $\sqrt[3]{x^2}$? And if "no", why?? This is something I've been struggling with since a while, for I got ambiguous answers.
Why $x^{2/3}$ has no graph in the negative $x$-axis? I mean isn't this $(x^2)^{1/3}$? then even if $x = -3$, $x^2$ makes it positive, or vice versa: $x^{1/3}$ is made positive by the successive square.
I am so stuck in this banality....
I'm guessing you asked your calculator to graph $x^{2/3}$ and it didn't show anything for negative $x$.
Non-integer powers of negative numbers are rather tricky. If the exponent is $a/b$ where $a$ and $b$ are integers and $b$ is odd, you can do it (and get a negative result if $a$ is odd, or a positive result if $a$ is even). But if $a$ is odd and $b$ is even, or if the exponent is irrational, there is no real solution (you can do it with complex numbers, but I suspect you're not yet at that level). So $x^{2/3}$ with $a=2$ even and $b=3$ odd should be ok.
However, calculators generally don't work with exact arithmetic, rather with decimal approximations. The calculator doesn't even have an exact value for $2/3$, it might use $0.666666666667$ (where the number of $6$'s depends on the calculator): that is literally $666666666667/1000000000000$. It's close to $2/3$, but not exactly the same, and it's a fraction with even denominator and odd numerator. When the calculator is asked to take a negative number to this power, it says "Oops: the exponent has an even denominator and odd numerator!" and refuses to return an answer.
Some calculators may (internally) use base $2$ rather than base $10$, but the end result is similar.