Without actually performing the long division, state whether the following rational number has terminating or non-terminating repeating (recurring) decimal expansion.$\frac{17}{8}$
Given :
The given rational number is $\frac{17}{8}$.
To do :
We have to find the decimal expansion of given rational number is terminating or
non-terminating repeating.
Solution :
The rational number $\frac{p}{q}$ is terminating, if,
i) p and q are coprime.
ii) q should be in the form of $2^n5^m$
In $\frac{17}{8}$,
17 and 8 has no common factors other than 1.
So, they are coprime.
Denominator $8 = 2\times 2\times 2 = 2^3$
It can be written as, $2^3 \times 5^0$
$\frac{17}{8} =\frac{17}{2^3 \times 5^0} $.
So, denominator is in the form of $2^n5^m$.
Therefore, the rational number $\frac{17}{8}$ has terminating decimal expansion.
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