I would like to be able to calculate the interest rate that is compounded for a given total interest rate, and number of compounding events.
TotalInterestRate = ((1+CompoundingRate/#CompoundingEvents)^#CompoundingEvents)-1
Basically would like to solve this equation for Compounding Rate.
For a more concrete example, calculating the Monthly Interest Rate from a given APY. The formula for APY given a Monthly Interest Rate would then be.
APY = ((1+InterestRate/12)^12)-1
What would be a good way to go about solving this equation?
If we refer to the effective interest rate as $r$ (what you call the total interest rate) and we refer to the nominal interest rate as $n$ (what you call the compounding rate) and the compounding frequency as $N$, the formula you have is $$r=\left(1+\frac{n}{N}\right)^N-1$$ To solve for $n$:
$$\begin{align} r+1 &=\left(1+\frac{n}{N}\right)^N\\ \sqrt[N]{r+1}&=1+\frac{n}{N}\\ \sqrt[N]{r+1}-1&=\frac{n}{N}\\ N\left(\sqrt[N]{r+1}-1\right)&=n\\ \end{align}$$