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If 2\(^{x + y}\) = 16 and 4\(^{x - y} = \frac{1}{32}\), find...

If 2\(^{x + y}\) = 16 and 4\(^{x - y} = \frac{1}{32}\), find the values of x and y.

A.

x = \(\frac{3}{4}\), y = \(\frac{11}{4}\)

B.

x = \(\frac{3}{4}\), y = \(\frac{13}{4}\)

C.

x = \(\frac{2}{3}\), y = \(\frac{4}{5}\)

D.

x = \(\frac{2}{3}\), y = \(\frac{13}{4}\)

Correct answer is B

2\(^{x + y}\) = 16 ; 4\(^{x - y}\) = \(\frac{1}{32}\).

\(\implies 2^{x + y} = 2^4\)

\(x + y = 4 ... (1)\)

\(2^{2(x - y)} = 2^{-5} \)

\(2^{2x - 2y} = 2^{-5}\)

\(\implies 2x - 2y = -5 ... (2)\)

Solving the equations (1) and (2) simultaneously, we have

x = \(\frac{3}{4}\) and y = \(\frac{13}{4}\)