If I have let's say $\displaystyle x^\frac{-1}{3}=2x$
If I take the reciprocal of the left hand side to eliminate the minus, do I have to do the same for both sides of the equation?
Yes. In an equation, whatever you do to one side, do it also to the other side so that the equality remains the same.
But why would you like to get the reciprocal of x^(-1/3) to make the -1/3 power become 1/3 power?
The reciprocal of x^(-1/3) is 1 / x ^(-1/3) = x^(1/3)
So you new equation is x^(1/3) = 1/(2x).
Why not just put the x^(-1/3) at the denominator to "eliminate the negative"?
x^(-1/3)
= 1 / x^(1/3)
So
1 / x^(1/3) = 2x.
Of course, whichever way,
2x * x^(1/3) = 1