Rather than getting yourself tangled up badly in exponents, write a simple Excel program. It's more interesting if you start by making n your variable (and n does not need to be an integer, just real and positive). Put the value for n^(1/n) in the first column all the way down. At the top the second column also put that value, and below it raise the value above by the value in the first colum. Copy that formula down the second column, and in fairly short order you will converge on n.

See is believing.