Hello. This is my first post and I'll readily admit I'm not a good mathematician; but now and then I get interested in something and try to work it out.
What I have noticed is that:
Sqrt (a^2 + b^2) > cube root (a^3 + b^3) > 4th root (a^4 + b^4)... etc
e.g. sqrt (3^2 + 4^2) = 5
which is greater than cube root (3^3 + 4^3) = 4.498
which is greater than fourth root (3^4 + 4^4) = 4.285
and so on.
However, is there any way to *prove* that in general:
nth root (a^n + b^n) > n+1th root (a^n+1 + b^n+1)?