problem: Simplify: $\displaystyle \frac{\frac 12\times 1024}{0.125\times 2^{12}}$.
solution: Each number in the expression equals 2 raised to an integer power, so we can use the laws of exponents to simplify the expression: \[\displaystyle \frac{\frac 12\times 1024}{0.125\times 2^{12}} = \frac{2^{-1} \times 2^{10}}{2^{-3}\times 2^{12}} = \frac{2^{-1+10}}{2^{-3+12}} = \frac{2^9}{2^9} = \boxed{1}.\]