1. The problem asks for the multiplicative inverse of $\frac{3}{2}$ raised to the power of $-4$ in exponential form. 2. The multiplicative inverse of a number $x$ is $\frac{1}{x}$. 3. So, the multiplicative inverse of $\frac{3}{2}$ is $\frac{2}{3}$. 4. Now, raise this inverse to the power of $-4$: $$\left(\frac{2}{3}\right)^{-4}$$ 5. Using the property of exponents $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$, we get: $$\left(\frac{2}{3}\right)^{-4} = \left(\frac{3}{2}\right)^4$$ 6. Calculate $\left(\frac{3}{2}\right)^4$: $$\left(\frac{3}{2}\right)^4 = \frac{3^4}{2^4} = \frac{81}{16}$$ 7. Therefore, the multiplicative inverse of $\frac{3}{2}$ raised to the power $-4$ is $\frac{81}{16}$. Final answer: $$\boxed{\frac{81}{16}}$$