Without actually performing the long division state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
$\frac{17}{8}$ .

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Solution

If the denominator has factors of the form of $2^{m} \times 5^{n}$ then it has a terminating decimal expansion.
If the denominator has factors other than $2$ and $5$ then it has a non-terminating decimal expansion.
Now, factorizing the denominator, we get,
$8 = 2 \times 2 \times 2 = 2^{3}$
As the denominator has only $2$ as its factor.
Hence $\frac{17}{8}$ has a terminating decimal expansion.

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