The decimal expansion of the rational number

Question:

The decimal expansion of the rational number $\frac{43}{2^{4} \times 5^{3}}$ will terminate after how many places of decimals?

Solution:

We have,

$\frac{43}{2^{4} \times 5^{3}}$

Theorem states: 

Let $x=\frac{p}{q}$ be a rational number, such that the prime factorization of $q$ is of the form $2^{m} \times 5^{n}$, where $m$ and $n$ are nonnegative integers.

Then, x has a decimal expression which terminates after k places of decimals, where k is the larger of m and n.

This is clear that the prime factorization of the denominator is of the form $2^{m} \times 5$

Hence, it has terminating decimal expansion which terminates after 4 places of decimal.

 

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