Express the following fraction in simplest form, only using positive exponents. \[ \frac{\left(-3 y^{-1}\right)^{3}}{-2 y^{-1}} \]

To simplify the fraction \(\frac{\left(-3 y^{-1}\right)^{3}}{-2 y^{-1}}\), we first simplify the numerator. Calculating the numerator: \[ \left(-3 y^{-1}\right)^{3} = (-3)^{3} \cdot (y^{-1})^{3} = -27 \cdot y^{-3} \] Thus, the numerator simplifies to: \[ -27 y^{-3} \] Now, we can rewrite the entire fraction: \[ \frac{-27 y^{-3}}{-2 y^{-1}} \] Since both the numerator and the denominator are negative, we can simplify: \[ \frac{27 y^{-3}}{2 y^{-1}} \] Next, we can simplify the fraction. When dividing by a common base, we subtract the exponents: \[ \frac{27}{2} \cdot \frac{y^{-3}}{y^{-1}} = \frac{27}{2} \cdot y^{-3 - (-1)} = \frac{27}{2} \cdot y^{-2} \] Now, writing \(y^{-2}\) with a positive exponent gives us: \[ \frac{27}{2} \cdot \frac{1}{y^{2}} = \frac{27}{2y^{2}} \] Finally, the expression in simplest form is: \[ \frac{27}{2y^{2}} \]