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If \((\frac{2}{3})^{m} (\frac{3}{4})^{n} = \frac{256}{729}\)...

If \((\frac{2}{3})^{m} (\frac{3}{4})^{n} = \frac{256}{729}\), find the values of m and n.

A.

m = 4, n = 2

B.

m = -4, n = -2

C.

m = -4, n =2

D.

m = 4, n = -2

E.

m = -2, n = 4

Correct answer is D

(\(\frac{2}{3}\))m (\(\frac{3}{4}\))n = \(\frac{256}{729}\)

\(\frac{2^m}{4^n}\) x \(\frac{3^n}{3^m}\) = \(\frac{2^{8}}{3^{6}}\)

\(2^{m} \div 2^{2n} = 2^{8}; 3^{n} \div 3^{m} = 3^{-6}\)

m - 2n = 8........(i)

-m + n = -6........(ii)

Solving the equations simultaneously

m = 4, n = -2