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 $\displaystyle n$ for which $\displaystyle 5^n+100$ is a perfect square
$\displaystyle 5^n+100=m^2$ for some $\displaystyle m$
assume $\displaystyle n\geq 3$. Now m must be divisible by 5 so we can write m=5t.
replacing we get
$\displaystyle 5^{n-2}=(t-2)(t+2)$
so $\displaystyle t-2$ and $\displaystyle t+2$ are both powers of 5 but cannot both be divisible by 5 since otherwise their difference = 4 would be divisible by 5
therefore $\displaystyle t-2=1$ and $\displaystyle t+2=5^{n-2}$
this gives t=3 and n=3