Simplify: \[ \frac{6\cdot8^{n+1} + 16\cdot2^{3n-2}}{10\cdot2^{3n-1} – 7\cdot8^n} \]
Solution
\[ 8 = 2^3 \]
\[ = \frac{6\cdot(2^3)^{n+1} + 16\cdot2^{3n-2}}{10\cdot2^{3n-1} – 7\cdot(2^3)^n} \]
\[ = \frac{6\cdot2^{3n+3} + 2^4\cdot2^{3n-2}}{10\cdot2^{3n-1} – 7\cdot2^{3n}} \]
\[ = \frac{2^{3n+2}(12 + 1)}{2^{3n-1}(10 – 14)} \]
\[ = \frac{13\cdot2^{3n+2}}{-4\cdot2^{3n-1}} \]
\[ = -13\cdot2^3 \]
\[ = -104 \]
Final Answer:
\[ \boxed{-104} \]