1. **Problem statement:** Calculate the value of $A = \frac{\left(-\frac{4}{3}\right)^4}{\left(\frac{4}{5}\right)^2} \times \left(-\frac{2}{3}\right)^{2^2}$ and express it as a power of a number. 2. **Formula and rules:** Recall that $a^{-n} = \frac{1}{a^n}$, $(a/b)^n = \frac{a^n}{b^n}$, and powers multiply when bases are the same. 3. **Calculate each part:** - $\left(-\frac{4}{3}\right)^4 = \left(\frac{4}{3}\right)^4 = \frac{4^4}{3^4} = \frac{256}{81}$ (even power removes negative sign) - $\left(\frac{4}{5}\right)^2 = \frac{16}{25}$ - $\left(-\frac{2}{3}\right)^{2^2} = \left(-\frac{2}{3}\right)^4 = \left(\frac{2}{3}\right)^4 = \frac{16}{81}$ 4. **Substitute and simplify:** $$A = \frac{\frac{256}{81}}{\frac{16}{25}} \times \frac{16}{81} = \frac{256}{81} \times \frac{25}{16} \times \frac{16}{81}$$ 5. **Cancel common factors:** $$= \frac{256}{\cancel{81}} \times \frac{25}{\cancel{16}} \times \frac{\cancel{16}}{81} = \frac{256 \times 25}{81 \times 81} = \frac{6400}{6561}$$ 6. **Express as power:** Note $6400 = 80^2$ and $6561 = 81^2$, so $$A = \left(\frac{80}{81}\right)^2$$ **Final answer:** $A = \left(\frac{80}{81}\right)^2$. --- **Summary:** - $A = \left(\frac{80}{81}\right)^2$