Write the denominator of the rational number $\frac{257}{5000}$ in the form $2^{m} \times 5^{n},$ where $m, n$ are non-negative integers. Hence,write its decimal expansion,without actual division.

(D) The denominator of the rational number $\frac{257}{5000}$ is $5000$.
First,we find the prime factorization of $5000$:
$5000 = 5 \times 1000 = 5 \times 10^3 = 5 \times (2 \times 5)^3 = 5 \times 2^3 \times 5^3 = 2^3 \times 5^4$.
This is in the form $2^m \times 5^n$,where $m = 3$ and $n = 4$ are non-negative integers.
To find the decimal expansion without actual division,we make the powers of $2$ and $5$ equal:
$\frac{257}{5000} = \frac{257}{2^3 \times 5^4}$.
To make the powers equal to $4$,we multiply the numerator and denominator by $2^1$:
$\frac{257 \times 2}{2^3 \times 5^4 \times 2^1} = \frac{514}{2^4 \times 5^4} = \frac{514}{(2 \times 5)^4} = \frac{514}{10^4}$.
$\frac{514}{10000} = 0.0514$.
Thus,the decimal expansion is $0.0514$.