HMMT-Nov 2011 guts, Problem 3

Let $x$ be a real number such that $2^{x}=3$. Determine the value of $4^{3 x+2}$.

Solution: We have \[ 4^{3 x+2}=4^{3 x} \cdot 4^{2}=\left(2^{2}\right)^{3 x} \cdot 16=2^{6 x} \cdot 16=\left(2^{x}\right)^{6} \cdot 16=3^{6} \cdot 16=11664 \] $\fbox{11664}$.

Answer: 11664