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7^(1/7) = m/n

then

7 = (m^7)/(n^7)

implies m^7 is a multiple of 7.

Means m^7 = 0 mod 7

So, using fermats little theorem..

m^7 = m mod 7

for m to be in the class of 0 it has to be a multiple of 7.

Now set m = 7k, so

7n^7 = 49k^7

But now I'm stuck, how do I show from here that m/n has a common factor?

Thanks