Answer:
Let's simplify the given equation and solve for \(x\):
\[ (25^{2x-1}) \cdot \left(\frac{1}{125}\right)^{x+2} = 625 \]
First, express \(25\) and \(\frac{1}{125}\) with the same base:
\[ (5^2)^{2x-1} \cdot (5^{-3})^{x+2} = 625 \]
Simplify the exponents:
\[ 5^{4x-2} \cdot 5^{-3x-6} = 625 \]
Combine the terms with the same base:
\[ 5^{4x-2 - 3x-6} = 625 \]
Simplify the exponent:
\[ 5^{x-8} = 625 \]
Now, express \(625\) as \(5^4\) and solve for \(x\):
\[ 5^{x-8} = 5^4 \]
Since the bases are the same, the exponents must be equal:
\[ x-8 = 4 \]
Solve for \(x\):
\[ x = 12 \]
So, the solution set is \( \{12\} \), and the correct answer is D. {12}.
