# What is the multiplicative inverse of - \frac{z^3}{2xy^2}?

The muplticative inverse of a number $x$ is, by definition, a number $y$ such that $x \setminus \cdot y = 1$.
So, in case of integer numbers $n$, the multiplicative inverse of $n$ is simply $\setminus \frac{1}{n}$, and thus it's not an integer number.
In the case of fractions, instead, the multiplicative inverse of a fraction is still a fraction, and it's simply a fraction with the same positivity of the original one, and with numerator and denominator flipped over: the multiplicative inverse of $\setminus \frac{a}{b}$ is the fraction $\setminus \frac{b}{a}$. So, in your case, the multiplicative inverse of $- \setminus \frac{{z}^{3}}{2 x {y}^{2}}$ is $- \setminus \frac{2 x {y}^{2}}{{z}^{3}}$.