Find n (natural number) where sqrt(1^{3}+3^{3}+3^{3}+4^{3}+5^{n}) is irrational.
in your other thread with the same problem, I just tried $n=1$ ... once again, on your edited expression I tried $n=1$ ...
$\sqrt{1^3+2^3+3^3+4^3+5^1} = \sqrt{105}$
As stated by Plato in your previous thread, note that 105 is not a perfect square, therefore its square root is irrational.
Let us find all for which is a perfect square
for some
assume . Now m must be divisible by 5 so we can write m=5t.
replacing we get
so and are both powers of 5 but cannot both be divisible by 5 since otherwise their difference = 4 would be divisible by 5
therefore and
this gives t=3 and n=3