First, we add 25 and 17:
\[ 25 + 17 = 42 \]
Next, we raise the sum to the 5th power:
\[ 42^5 \]
Now calculating \(42^5\):
\[ 42^5 = 42 \times 42 \times 42 \times 42 \times 42 \]
Calculating it step-by-step:
\[ 42^2 = 1764 \]
\[ 42^3 = 42 \times 1764 = 74088 \]
\[ 42^4 = 42 \times 74088 = 3111696 \]
\[ 42^5 = 42 \times 3111696 = 130691232 \]
Thus,
\[ 42^5 = 130691232 \]
The final answer is:
\[ \boxed{130691232} \]