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De Moivre's Theorem

De Moivre's Theorem shows that to raise a complex number to the nth power, the absolute value is raised to the nth power and the argument is multiplied by n.

Level 3

         

\[\large { z }^{ 3 }=1\]

What is the set of all \(z\) that satisfy the equation above?

Note: \(\omega = \frac{ -1 + \sqrt 3 i}{2} \) where \(i =\sqrt{-1} \).

Find the value of \[(2-\omega)(2-\omega^2)(2-\omega^{10})(2-\omega^{11}).\]

Details and Assumptions:

\(\omega\) is a non-real cube root of unity.

The five roots of the equation \(z^{5}=4-4i\) each take the form

\[\Large \sqrt{2} e ^ { \frac{ k \pi i } { 20} }, \]

where \(k\) is a positive integer less than 40.

Find the sum of all values of \(k\).

\[\large x + \frac 1 x = \sqrt 3\ , \ \ \ \ \ \ \ x^{200} + \frac {1}{x^{200}} = \ ? \]

If the \(6\) solutions of \(x^{6}=-64\) are written in the form \(a+ib\), where \(a\) and \(b\) are real, then what is the product of those solutions with \(a>0\)?

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