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

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

View answer & explanation

Correct answer is A

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

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

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

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

Equating bases, we have

$2 - 2x + x + 2 + 3x = -4$

$4 + 2x = -4 \implies 2x = -4 - 4$

$2x = -8$

$x = -4$

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