Algebra

De Moivre's Theorem

De Moivre's Theorem: Level 3 Challenges

         

z3=1\large { z }^{ 3 }=1

What is the set of all zz that satisfy the equation above?

Note: ω=1+3i2\omega = \frac{ -1 + \sqrt 3 i}{2} where i=1i =\sqrt{-1} .

Find the value of (2ω)(2ω2)(2ω10)(2ω11).(2-\omega)(2-\omega^2)(2-\omega^{10})(2-\omega^{11}).

Details and Assumptions:

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

The five roots of the equation z5=44iz^{5}=4-4i each take the form

2ekπi20,\Large \sqrt{2} e ^ { \frac{ k \pi i } { 20} },

where kk is a positive integer less than 40.

Find the sum of all values of kk.

x+1x=3 ,       x200+1x200= ?\large x + \frac 1 x = \sqrt 3\ , \ \ \ \ \ \ \ x^{200} + \frac {1}{x^{200}} = \ ?

If the 66 solutions of x6=64x^{6}=-64 are written in the form a+iba+ib, where aa and bb are real, then what is the product of those solutions with a>0a>0?

×

Problem Loading...

Note Loading...

Set Loading...