jmc

The equation, $(2^{x+1}{)}^3\cdot 4^x=8192$, can be written as $2^{3x+3}\cdot 4^x=8192$. We also know that $2^{3x+3}=2^{3x}\cdot 2^3$ and $4^x=2^{2x}$. Using substitution we have, $2^{3x}\cdot 2^3\cdot 2^{2x}=8192$. Next, we combine like terms to get $2^{5x}\cdot 8=8192$. After dividing both sides of the equation by $8$, we find that $2^{5x}=1024$. Since $1024=2^{10}$, $2^{5x}=2^{10}$ and $x=\boxed{2}$.