Mathematica, 3535 40
n /. Solve[Nest[#^(1/n) &, a, b] == n]~N~5
Generates a list of all solutions, with 5 digit precision.
n /. Last@Solve[Nest[#^(1/n) &, a, b] == n]~N~5
5 more characters to get only the real solution, which the updated rules demand.