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There are four different fourth roots of 16. That is, there are four complex numbers that can be x in the equation below:

x^4=16 

What are the four complex numbers?

 Sep 25, 2019
edited by Guest  Sep 25, 2019

Best Answer 

 #1
avatar+6046 
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\(16 = 16 e^{i 2\pi k},~k \in \mathbb{Z}\\ \sqrt[4]{16} = \sqrt[4]{16}e^{i \frac{2\pi k }{4}},~k = 0, 1, 2, 3 = \\ 2e^{0},~2e^{i\pi/2},~2e^{i\pi},~2e^{i3\pi/2} = \\ 2, 2i, -2, -2i\)

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 Sep 25, 2019
 #1
avatar+6046 
+1
Best Answer

\(16 = 16 e^{i 2\pi k},~k \in \mathbb{Z}\\ \sqrt[4]{16} = \sqrt[4]{16}e^{i \frac{2\pi k }{4}},~k = 0, 1, 2, 3 = \\ 2e^{0},~2e^{i\pi/2},~2e^{i\pi},~2e^{i3\pi/2} = \\ 2, 2i, -2, -2i\)

Rom Sep 25, 2019
 #2
avatar+2557 
+1

I haven't learned this yet, so just curious, how would 2 be considered to be a complex number?

CalculatorUser  Sep 26, 2019
 #3
avatar+6046 
+1

All reals are also complex numbers.

 

A complex number need not have an imaginary part.

Rom  Sep 26, 2019

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