Error raising a complex number to a power

Question

I am trying to do

$(3+7i)^5$

which acording to WolframAlpha and Mathway should be:

$23028−11228i$

Yet I instead get: $6123+14287i$

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I'm getting that answer by doing:

$3^5 (7i)^0 + 3^4 (7i)^1 + 3^3 (7i)^2 + 3^2 (7i)^3 + 3^1 (7i)^4 + 3^0 (7i)^5$

and then just combining the like terms. Could anyone shed some light on what I'm doing wrong??

Thanks in advance!!

Answer 1

As John pointed out in the comment, you have used $$ (a+b)^n = \sum_{k=0}^n a^k b^{n-k}, $$ which is wrong. Instead, you should use $$ (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^k b^{n-k}. $$

Answer 2

Hint: An easier way to do this might be to first convert your number into polar form: $z = re^{i\theta}$. Then $z^5 = re^{i5\theta}$. From there, you can re-expand using Euler's formula.

Answer 3

You can save time by doing it another way:
By Euler :$3+7i=e^{iarctan\frac 73}$,
which means $(3+7i)^5=e^{i5arctan\frac 73}$
Now use a calculator to evaluate $5arctan \frac 73$. If we know that $5arctan \frac 73=\phi$
Then by Euler's equation again: $(3+7i)^5=e^{i\phi}=cos\phi+isin\phi$ . As to the reason why you didn't get it right:
$3^5 (7i)^0 + 3^4 (7i)^1 + 3^3 (7i)^2 + 3^2 (7i)^3 + 3^1 (7i)^4 + 3^0 (7i)^5$should actually be:
$3^5 (7i)^0 + 5*3^4 (7i)^1 + 10*3^3 (7i)^2 + 10*3^2 (7i)^3 + 5*3^1 (7i)^4 + 3^0 (7i)^5$.
To calculate the coefficients: Suppose we want to expand $(a+b)^n$, then $(a+b)^n=\Sigma^n_{r=0} \frac{n!}{r!(n-r)!}a^rb^{n-r}$. Now you get it.