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 implies 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 $5^{10}$. Therefore, there are $\boxed{10}$ digits to the right of the decimal point in $\frac{1}{2^{10}}$.