1. **Problem Statement:** Given matrix $$A = \begin{bmatrix}-7 & -9 \\ 6 & 8\end{bmatrix}$$ and its diagonalization $$A = P D P^{-1}$$, express $$A^5$$ in terms of $$P$$, a power of $$D$$, and $$P^{-1}$$. 2. **Recall the diagonalization property:** If $$A = P D P^{-1}$$ where $$D$$ is a diagonal matrix, then powers of $$A$$ can be computed as $$A^n = P D^n P^{-1}$$ for any positive integer $$n$$. 3. **Apply the property for $$n=5$$:** $$$ A^5 = P D^5 P^{-1} $$$ 4. **Explanation:** Since $$D$$ is diagonal, raising it to the 5th power means raising each diagonal element to the 5th power. This simplifies computation of $$A^5$$ without directly multiplying $$A$$ five times. **Final answer:** $$$ A^5 = P D^5 P^{-1} $$$