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# Write the denominator of rational number $\frac{257}{5000}$ in the form $2^m \times5^n$,where $m, n$ are non-negative integers. Hence, write its decimal expansion, without actual division

Given:

The given rational number is $\frac{257}{5000}$.

To do:

Here, we have to write the decimal expansion of the given rational number by writing its denominator in the form of $2^m \times 5^n$, where m, and n, are non-negative integers.

Solution:

$\frac{257}{5000}=\frac{257}{5\times1000}=\frac{257}{5\times2^3\times5^3}=\frac{257}{2^3\times5^4}$

We can see that $2^3\times5^4$ is of the form $2^m \times 5^n$, where $m =3$ and $n = 4$.

This implies,

The given rational number has a terminating decimal expansion.

Multiply the numerator and denominator by $2^1$ so that the denominator becomes a multiple of $10^r$, where r is any positive integer.

Therefore,

$\frac{257}{2^3\times5^4}=\frac{257\times2^1}{2^1\times2^3\times5^4}$

$=\frac{257\times2}{(2\times5)^4}$

$=\frac{514}{10^4}$

$=\frac{514}{10000}$

$=0.0514$

The decimal expansion of the given rational number is $0.0514$.

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