If \(25^{1 - x} \times 5^{x + 2} \div (\frac{1}{125})^{x} = 625^{-1}\), find the value of x.
A. x = -4
B. x = 2
C. x = -2
D. x = 4
Correct Answer: A
Explanation
\(25^{1 - x} \times 5^{x + 2} \div (\frac{1}{125})^{x} = 625^{-1}\)
\((5^2)^{(1 - x)} \times 5^{(x + 2)} \div (5^{-3})^x = (5^4)^{-1}\)
\(5^{2 - 2x} \times 5^{x + 2} \div 5^{-3x} = 5^{-4}\)
\(5^{(2 - 2x) + (x + 2) - (-3x)} = 5^{-4}\)
Equating bases, we have
\(2 - 2x + x + 2 + 3x = -4\)
\(4 + 2x = -4 \implies 2x = -4 - 4\)
\(2x = -8\)
\(x = -4\)
