If \(25^{1 - x} \times 5^{x + 2} \div (\frac{1}{125})^{x} = 625^{-1}\), find the value of...

If \(25^{1 - x} \times 5^{x + 2} \div (\frac{1}{125})^{x} = 625^{-1}\), find the value of x.

\(25^{1 - x} \times 5^{x + 2} \div (\frac{1}{125})^{x} = 625^{-1}\)

\((5^2)^{(1 - x)} \times 5^{(x + 2)} \div (5^{-3})^x = (5^4)^{-1}\)

\(5^{2 - 2x} \times 5^{x + 2} \div 5^{-3x} = 5^{-4}\)

\(5^{(2 - 2x) + (x + 2) - (-3x)} = 5^{-4}\)

Equating bases, we have

\(2 - 2x + x + 2 + 3x = -4\)

\(4 + 2x = -4 \implies 2x = -4 - 4\)

\(2x = -8\)

\(x = -4\)

There is an explanation video available below.

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