Question

Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

\[\frac{129}{2^2 \times 5^7 \times 7^{17}}\]

Answer 1

 The given number is \[\frac{129}{2^2 \times 5^7 \times 7^{17}}\]

Clearly, none of 2, 5 and 7 is a factor of 129.

So, the given number is in its simplest form.

Now,

\[2^2 \times 5^7 \times 7^{17}\]  is not of the form

\[2^m \times 5^n\] 

So, the given number has a non-terminating repeating decimal expansion.