Effective Annual Rate from a Nominal Rate

A quoted rate is not a cost until you know how often it compounds. "12% a year" charged monthly is really 1% twelve times, and $(1.01)^{12} - 1 = 12.68\%$. The extra 0.68 points is interest earned on interest, and it grows with the compounding frequency: the same 12% compounded daily comes to 12.747%, approaching the continuous limit $e^{0.12} - 1 = 12.75\%$.

This is the only fair way to compare two offers. A card at 19.99% compounded daily and a line of credit at 20.2% compounded annually are not what they appear, and the nominal figures rank them the wrong way round. Disclosure law exists precisely because of this gap, which is why lenders must publish an effective or annualised figure alongside the headline rate.

Going backwards recovers the nominal rate a lender must be quoting to produce a given effective one. Notice that $m$ cannot be solved for: it sits in the base and the exponent at once, and no elementary rearrangement frees it.