Has the rational number


Has the rational number $\frac{441}{2^{2} \times 5^{7} \times 7^{2}}$ a terminating or a nonterminating decimal representation?


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.

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