Multiplying by Inverse of a Fraction question

Question

Hey so I'm doing an exercise and I got a bit confused by something.

I've learned early on that you can multiply the inverse of a fraction and would get the same result as you would if you divided, since you do the exact opposite.

Then why doesn't this hold true for:

$$\frac{10z^{1/3}} {2z^{2}}{^{}{}} = {10z^{1/3}} * {2z^{-2}}$$

Please explain in simple words.. not too math savvy ^^

Answer 1

The reason is that you forgot to use the inverse of $2$ as well. So it should be $$10z^{1/3}\cdot 2^{-1}z^{-2}$$ That's because in the original expression, you are dividing by $2$ and also by $z^2$.

In other words, $(2z^2)^{-1}$ can be written as $2^{-1}z^{-2}$, but not as $2z^{-2}$.

Answer 2

The problem is the $2$. What you should do is to take the whole denominator (the thing you divide with) to $^{-1}$. Thus you should get $$\frac{10z^{1/3}}{2z^2} = 10z^{1/3}*(2z^{2})^{-1} = 10z^{1/3}*2^{-1}*z^{-2} =10z^{1/3}*\frac{z^{-2}}{2} $$