SOLUTION: Evaluate {{{ ( 1 + i )^12 }}} by using De Moivre's Theorem. Express the result in rectangular form. Thank you!

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Question 238652: Evaluate +%28+1+%2B+i+%29%5E12+ by using De Moivre's Theorem. Express the result in rectangular form.
Thank you!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First you have to convert into polar form .

To do that, use the formulas and


In this case, we're given which means that x=1 and y=1. So and

So this means that the rectangular expression z=1%2Bi is equivalent to the polar form .

From here, we can now use De Moivre's Theorem. De Moivre's Theorem states that if , then

Since , using De Moivre's Theorem gets us






... Evaluate sqrt%282%29 to the 12th power to get %28sqrt%282%29%29%5E12=%282%29%5E%28%281%2F2%29%2812%29%29=2%5E6


... Evaluate 2 to the 6th power to get 64


... Evaluate the trig functions.


... Simplify.


... Multiply


Because we let z=1%2Bi, this means that %281%2Bi%29%5E12=-64