Why is the graph of $f(x) = (x^2)^\frac{1}{6}$ not the same graph as $f(x) = x^\frac{1}{3}$?
Shouldn't they be the same because when you apply the exponent rules to the first equation, you get the same result as the second equation?
By the way, I am an Algebra 2 student in High School.
When you square the quantity $x$, the negative sign disappears, and it doesn't come back if you raise it to the $\frac{1}{6}$ power. If you just take $\frac{1}{3}$ power, then the negative stays negative.
$((-1)^2)^\frac{1}{6}=1^\frac{1}{6}=1$, while $(-1)^\frac{1}{3}=-1$
Therefore, what you see on the first graph is the absolute value of the second graph.
This hyperlink contains a graph of the two equations. Note that when $x<0$ the red solid line is the blue dotted line reflected about the x-axis.