1. The problem involves evaluating the expression $$\left| x+3 \begin{bmatrix}a&b\\c&d\end{bmatrix}^n \right|$$ where the matrix $$\begin{bmatrix}a&b\\c&d\end{bmatrix}$$ is raised to the power $$n$$ and multiplied by $$x+3$$, then the absolute value is taken. 2. To solve this, we need to understand matrix powers and determinants. The determinant of a product of matrices equals the product of their determinants: $$|AB| = |A||B|$$. 3. The determinant of a scalar times a matrix is $$|\lambda A| = |\lambda|^m |A|$$ where $$m$$ is the size of the square matrix (here $$m=2$$). 4. Therefore, the determinant of $$x+3$$ times the matrix power is: $$ \left| (x+3) \begin{bmatrix}a&b\\c&d\end{bmatrix}^n \right| = |x+3|^2 \cdot \left| \begin{bmatrix}a&b\\c&d\end{bmatrix}^n \right| $$ 5. Since $$\left| M^n \right| = \left| M \right|^n$$ for any square matrix $$M$$, we have: $$ \left| \begin{bmatrix}a&b\\c&d\end{bmatrix}^n \right| = \left| \begin{bmatrix}a&b\\c&d\end{bmatrix} \right|^n = (ad - bc)^n $$ 6. Putting it all together: $$ \left| x+3 \begin{bmatrix}a&b\\c&d\end{bmatrix}^n \right| = |x+3|^2 (ad - bc)^n $$ 7. This formula allows you to compute the determinant for any $$n$$, given $$a,b,c,d$$ and $$x$$. Final answer: $$ \boxed{\left| x+3 \begin{bmatrix}a&b\\c&d\end{bmatrix}^n \right| = |x+3|^2 (ad - bc)^n} $$