How can I prove by mathematical induction that $$\sum_{i=0}^n i^4 = (\sum_{i=0}^n i)^3$$ ?
I see easily that it holds for $i=0$.
Using the inductive hypothesis, I get: $$\sum_{i=0}^{n+1} i^4 = (\sum_{i=0}^n i)^3 + (n+1)^4$$. I've been thinking about applying $\sum_{i=0}^n i = n(n+1)/2$, but this doesn't help me much here.
Should I expand all of this using the binomial theorem ?
What you're trying to prove isn't true. Counterexample: $n=2$. Then you get $$0^4 + 1^4 + 2^4 = 17 \neq (0+1+2)^3$$