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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Updated on: 10-Oct-2022

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