\( 1 \leftarrow \quad \) Simplify the following expression. Write the result using positive exponents. \[ \frac{\left(-3 x y^{-4}\right)^{-4}}{\left(x y^{-3}\right)^{-3}} \]

To simplify the expression \(\frac{\left(-3 x y^{-4}\right)^{-4}}{\left(x y^{-3}\right)^{-3}}\), we start by simplifying the numerator and the denominator separately. 1. **Numerator:** \[ \left(-3 x y^{-4}\right)^{-4} = \frac{1}{\left(-3 x y^{-4}\right)^{4}} = \frac{1}{(-3)^{4} (x)^{4} (y^{-4})^{4}} = \frac{1}{81 x^4 y^{-16}} = \frac{y^{16}}{81 x^4} \] 2. **Denominator:** \[ \left(x y^{-3}\right)^{-3} = \frac{1}{\left(x y^{-3}\right)^{3}} = \frac{1}{(x)^{3} (y^{-3})^{3}} = \frac{1}{x^3 y^{-9}} = \frac{y^{9}}{x^3} \] Now substitute these back into the expression: \[ \frac{\frac{y^{16}}{81 x^{4}}}{\frac{y^{9}}{x^{3}}} = \frac{y^{16}}{81 x^{4}} \cdot \frac{x^{3}}{y^{9}} = \frac{y^{16} x^{3}}{81 x^{4} y^{9}} \] Next, we simplify the fraction: \[ = \frac{x^{3}}{81 x^{4}} \cdot \frac{y^{16}}{y^{9}} = \frac{1}{81 x^{4-3}} \cdot y^{16-9} = \frac{1}{81 x} y^{7} \] Thus, the final simplified expression using positive exponents is: \[ \frac{y^{7}}{81 x} \]