Ternary expansion of fractions like 5/3³

[jakey]

Hi guys, I'd like to ask about ternary expansions. they seem easy but I'm having a hard time doing this as well as searching for tips online, specifically for $$x \in [0,1]$$.

I know that ternary expansions are similar to decimal expansions but for example, how do you find the ternary expansion of $$\frac{5}{3^3}$$ ?

 

[HallsofIvy]

In base 3, 5= 3+2 is $12_5$. Of course, $9= 3^2$ is $200_3$ so the fraction 5/9, in base 3, would be the fraction $12/200$. Can you write that in "trinary"?

 

[jakey]

Hi hallsofivy, sorry I didnt quite understand your reply.

And oh, it's 5/27 not 5/9. Perhaps a better way of explaining it to me would be how to identify the coefficients $$a_k$$ such that $$\frac{5}{27} = \sum \frac{a_k}{3^k}$$ ?