Question:
Has the rational number $\frac{441}{2^{2} \times 5^{7} \times 7^{2}}$ a terminating or a nonterminating decimal representation?
Solution:
We have,
$\frac{441}{2^{2} \times 5^{7} \times 7^{2}}$
Theorem states:
Let $x=\frac{p}{q}$ be a rational number, such that the prime factorization of $q$ is not of the form $2^{m} \times 5^{n}$, where $m$ and $n$ are nonnegative integers.
Then, x has a decimal expression which is non-terminating repeating.
This is clear that the prime factorization of the denominator is not of the form $2^{m} \times 5^{n}$.
Hence, it has non-terminating decimal expansion.