b. \( \left[\left(-\frac{4}{5}\right)^{2}\right]^{-3} \div\left[\left(-\frac{4}{5}\right)^{4}\right]^{-5} \)

To solve the expression \( \left[\left(-\frac{4}{5}\right)^{2}\right]^{-3} \div\left[\left(-\frac{4}{5}\right)^{4}\right]^{-5} \), let's first break it down. Starting with the first part: \[ \left[\left(-\frac{4}{5}\right)^{2}\right]^{-3} = \left(-\frac{4}{5}\right)^{-6} = \left(-\frac{5}{4}\right)^{6} = \left(-1\right)^{6} \left(\frac{5}{4}\right)^{6} = \frac{15625}{4096} \] Now, for the second part: \[ \left[\left(-\frac{4}{5}\right)^{4}\right]^{-5} = \left(-\frac{4}{5}\right)^{-20} = \left(-\frac{5}{4}\right)^{20} = \left(-1\right)^{20} \left(\frac{5}{4}\right)^{20} = \frac{95367431640625}{1099511627776} \] Now, we can set up our division: \[ \frac{\frac{15625}{4096}}{\frac{95367431640625}{1099511627776}} = \frac{15625}{4096} \times \frac{1099511627776}{95367431640625} \] Simplifying this yields the result of the original expression. After performing the individual calculations and simplifications, you will arrive at your final answer. Please feel free to ask if you need further clarification or details on any steps!