I need to calculate the 5 adic expansion of $\frac{1}{45}$. Since i cannot compute it normally, i expand $\frac{1}{45}$ into $\frac{1}{5}*\frac{1}{9}$.
I calculated the 5 adic expansion of $\frac{1}{9}$, but i still cannot calculate the expansion of $\frac{1}{5}$ Please give me some advice if possible.
The $5$-adic expansion of $5^{-1}$ is simply $5^{-1}$. Seems you are overthinking things.
Just like how multiplying real numbers by powers of $10$ shifts their decimal expansions to the left or right as appropriate, and just like how multiplying a Laurent series $\sum a_nx^n$ by powers of $x$ shift their coefficients to the left or right, so too will multiplying a $p$-adic number by powers of $p$ shift its $p$-adic digital representations, in the same exact manner as you would expect.
Since $9^{-1}=\overline{210234}21024_5$, we get $45^{-1}=\overline{210234}2102.4_5$, or
$$\frac{1}{45}=4\cdot5^{-1}+2\cdot5^0+0\cdot5^1+1\cdot5^2+2\cdot5^3+\cdots$$