Prove that $3^n>n^4$ if $n\geq8$ Proving that $3^n>n^4$ if $n\geq8$
I tried mathematical induction start from $n=8$ as the base case, but I'm stuck when I have to use the fact that the statement is true for $n=k$ to prove $n=k+1$.  Any ideas?
Thanks!

It can be proved by induction. The n+1 can be solved in the following way (assuming the claim is correct for n): need to be prove $3^{n+1}\geq (n+1)^4$. but we assumed that $3^n\geq n^4$ so the claim also can be written as $3*3^n\geq3*n^4\geq(n+1)^4$ but taking the 4th square on each side (both sides are non-negative), $3^\frac{1}{4}n\geq n+1\rightarrow 0.31n\geq 1$ which is correct since we prove the claim for $n\geq 8$.
