SOLUTION: Find (1 / sq. rt. 2 + i / sq. rt. 2) ^ 8 a.) -8 b.) 256 c.) 8 d.) 1 e.) None of these

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Question 122065: Find (1 / sq. rt. 2 + i / sq. rt. 2) ^ 8

a.) -8
b.) 256
c.) 8
d.) 1
e.) None of these
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%281%2Fsqrt%282%29%2Bi%2Fsqrt%282%29%29%5E8 Start with the given expression


It's very useful to note that cos%28pi%2F4%29=1%2Fsqrt%282%29 and sin%28pi%2F4%29=1%2Fsqrt%282%29 . So the expression is equivalent to


%28cos%28pi%2F4%29%2Bi%2Asin%28pi%2F4%29%29%5E8


Now we're going to use De Moivre's theorem to solve this problem.

Remember, De Moivre's theorem states: %28cos%28x%29%2Bi%2Asin%28x%29%29%5En=cos%28n%2Ax%29%2Bi%2Asin%28n%2Ax%29

So using De Moivre's theorem we get


cos%288%2Api%2F4%29%2Bi%2Asin%288%2Api%2F4%29



cos%282pi%29%2Bi%2Asin%282pi%29 Multiply 8 and pi%2F4 to get 8pi%2F4. Now reduce to get 2pi



1%2B0i Take the cosine of 2pi to get 1. Take the sine of 2pi to get 0.


1 Remove the zero term


So %281%2Fsqrt%282%29%2Bi%2Fsqrt%282%29%29%5E8 simplifies to 1.

In other words, %281%2Fsqrt%282%29%2Bi%2Fsqrt%282%29%29%5E8=1



So the answer is D)