Question 12 Marks
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.
Answer
View full question & answer→Denominator of the rationaol number $\frac{257}{5000}$ is 5000.
Now, factors of $5000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5 = (2)^3 \times (5)^4$, which is of the type $2^m \times 5^n$, where $m = 3$ and $n = 4$ are non-negative integers.
$\therefore$ Rational number $=\frac{257}{5000}=\frac{257}{2^3\times5^4}\times\frac{2}{2}$
[since, miltiplying numerator and denominater by 2]
$=\frac{514}{2^4\times5^4}=\frac{514}{(10)^4}$
$=\frac{514}{10000}=0.0514$
Hence, whcih is the requaired decimal expansion of the rational $\frac{257}{5000}$ and it is also a $5000$ terminating decimal number.
Now, factors of $5000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5 = (2)^3 \times (5)^4$, which is of the type $2^m \times 5^n$, where $m = 3$ and $n = 4$ are non-negative integers.
$\therefore$ Rational number $=\frac{257}{5000}=\frac{257}{2^3\times5^4}\times\frac{2}{2}$
[since, miltiplying numerator and denominater by 2]
$=\frac{514}{2^4\times5^4}=\frac{514}{(10)^4}$
$=\frac{514}{10000}=0.0514$
Hence, whcih is the requaired decimal expansion of the rational $\frac{257}{5000}$ and it is also a $5000$ terminating decimal number.