Question

find the power of the complex number
(3+i)^4

Answer 1

4

Answer 2

it would be 3^4 + i^4

Answer 3

Use DeMoivre's theorem. First convert \(3+i\) to polar form.

Answer 4

Yes its 4

Answer 5

You end up with 81 + i^4

Answer 6

@mkgarnett Then I suppose \((1+1)^4=2\), because \((1+1)^4=1^4+1^4\) ?

Answer 7

well i^2 equals 1 @SithsAndGiggles

Answer 8

my bad it equals -1

Answer 9

I'm not arguing over what the value of \(i^2\) is. I'm pointing out your mistaken reasoning.
\[(a+b)^n\not=a^n+b^n\]

Answer 10

thats how you do it though.. (a+b)\[\left( a +b \right)^{2} \] equals \[a ^{2} + b ^{2}\]

Answer 11

or you could even do ab^2

Answer 12

That's simply not true.
\[(a+b)^2=a^2+2ab+b^2\]
What you say is true only when one or both of \(a,b\) are 0.

Answer 13

i thought it was if they dont equal 0

Answer 14

youre using the foil method

Answer 15

@Emerson85 convert the given complex number of the form \(x+iy\) into the polar form, \(\large re^{i\theta}\) using
\[\begin{cases}
x^2+y^2=r^2\\
\tan\theta=\dfrac{y}{x}
\end{cases}\]
One you have that, use DeMoivre's theorem to raise to the fourth power:
\[\large\left(re^{i\theta}\right)^4=r^4e^{4i\theta}=r^4\bigg(\cos4\theta+i\sin4\theta\bigg)\]