How to calcuate $(a+bi)^n$ without using the De Moivre's formula?

Question

Sometimes, I need to calculate the value of $(a+bi)^n$ but without using De Moivre's formula. Is there a method to do that and what is its name?? thanks

Answer 1

You can use the binomial theorem, $$\sum_{k=0}^{n} \binom{n}{k} a^k (ib)^{n-k} = (a+ib)^n$$