+296 When fractions are expressed in different bases, they can be terminating or repeating. For example, when $\frac{1}{5}$ is expressed in base $3,$ the result is
\[0.\overline{0121}_3 = 0.01210121 \dots,\]
which is repeating.
When $\frac{1}{288}$ is expressed in base $19,$ is it terminating or repeating?
+1804 First, let's convert the numbers in to base 19.
We have
\((F3)_{19} = (15 × 19^1) + (3 × 19^0) = (288)_{10 }\)
This means that \(\frac{1}{288}_{10}=\frac{1}{F3}_{19}\)
Doing some base division, we have
\(\frac{1}{F3}_{19}=\overline{0.014F9AE6}\)
Thus, it is a repeating decimal.
So our answer is repeating.
Thanks! :)
