Having looked through my entire lecture notes section entitled p-adic numbers, I have concluded that there is not one mention of the word adic in the whole section. Hence, I have no idea how I am supposed to do this.

Calculate 1/3 as a 5-adic number

After wikipediaing, I have realized that it should look something like...

$\displaystyle a_0 + a_1 5 + a_2 + 5^2 + a_3 5^3 + \dots$

But I have no idea how to get there.

Things we did cover in the section were,

Principle of Domination

Solving $\displaystyle x^2 \equiv a(\mod p^n)$

Non Archemedean Values

Are any of these going to help with this question? I can't see how.