Simplify the expression. Write the answer with positive exponents only. Assume all variables represent nonzero real numbers. \[ \frac{\left(2 s^{5} t_{*}\right)^{-4}}{\left(s^{4} t^{6}\right)^{-5}}=\square \]

To simplify the expression \( \frac{\left(2 s^{5} t_{*}\right)^{-4}}{\left(s^{4} t^{6}\right)^{-5}} \), we start by simplifying the numerator and the denominator separately. First, simplify the numerator: \[ \left(2 s^{5} t_{*}\right)^{-4} = \frac{1}{(2 s^{5} t_{*})^{4}} = \frac{1}{2^{4} (s^{5})^{4} (t_{*})^{4}} = \frac{1}{16 s^{20} t_{*}^{4}} \] Now, simplify the denominator: \[ \left(s^{4} t^{6}\right)^{-5} = \frac{1}{(s^{4} t^{6})^{5}} = \frac{1}{(s^{4})^{5} (t^{6})^{5}} = \frac{1}{s^{20} t^{30}} \] Now we can combine the results: \[ \frac{\frac{1}{16 s^{20} t_{*}^{4}}}{\frac{1}{s^{20} t^{30}}} = \frac{1}{16 s^{20} t_{*}^{4}} \cdot \frac{s^{20} t^{30}}{1} = \frac{s^{20} t^{30}}{16 s^{20} t_{*}^{4}} \] Now we can simplify by cancelling \(s^{20}\): \[ = \frac{t^{30}}{16 t_{*}^{4}} \] Finally, expressing the result with positive exponents only gives us: \[ \frac{t^{30}}{16 t_{*}^{4}} \] Thus, the simplified expression is: \[ \frac{t^{30}}{16 t_{*}^{4}} \]