jmc

Multiply numerator and denominator of $\frac{1}{2^{10}}$ by $5^{10}$ to see that $\frac{1}{2^{10}}$ is equal to $\frac{5^{10}}{10^{10}}$. This shows that the decimal representation of $\frac{1}{2^{10}}$ is obtained by moving the decimal point ten places to the left in the decimal representation of $5^{10}$. Since $5^{10}$ has a units digit of 5 (as does every positive integer power of 5), we find that the last digit in the decimal expansion of $\frac{1}{2^{10}}$ is $\boxed{5}$.