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
Verified Correct Answer: D.
Step-by-Step Mathematical Solution
(\(\frac{2}{3}\))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
Source: Myschool Past Questions Archive (Record #32469)