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The initial problem to solve is:
ReplyDelete[(6^7+6^7+6^7+6^7+6^7)/(5^7+5^7+5^7+5^7+5^7+5^7+5^7)]^(1/6)
Right away, we see the numerator can be rewritten as
5(6^7), and the denominator becomes
6(5^7)
So the problem becomes
[5(6^7)/6(5^7)]^1/6
Inside the fraction, the 5 on top is canceled by the 5^7 on the bottom, and the 6 on the bottom is canceled by the 6^7 on the top, leaving you with
(6^6/5^6)^(1/6)
So the sixth root of (6^6/5^6) is just
6/5.
QED.