SOLUTION: How do you final all complex solutions and write all answers in both trigonometric and rectangular form? {{{x^3-8=0}}}

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Question 1116764: How do you final all complex solutions and write all answers in both trigonometric and rectangular form?

Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
.
Equation   = 0  has three complex solutions, that are the roots of degree 3 of 8:


            trigonometric form             complex number form  
            r*(cos(t) + i*sin(t))          a + bi


1)  z = 2 = 2*(cos(0°)   + i*sin(0°)    =  2 + i*0;



2)  z =     2*(cos(120°) + i*sin(120°)) =  2*((-1/2) + i*(sqrt(3)/2)}}} = -1 + i*sqrt(3);



3)  z =     2*(cos(240°) + i*sin(240°)) =  2*((-1/2) - i*(sqrt(3)/2)}}} = -1 - i*sqrt(3);

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On complex numbers, see the lessons in this site
    - Complex numbers and arithmetical operations on them
    - Complex plane
    - Addition and subtraction of complex numbers in complex plane
    - Multiplication and division of complex numbers in complex plane
    - Raising a complex number to an integer power
    - How to take a root of a complex number (*)
    - Solution of the quadratic equation with real coefficients on complex domain
    - How to take a square root of a complex number
    - Solution of the quadratic equation with complex coefficients on complex domain

    - Solved problems on taking roots of complex numbers (*)
    - Solved problems on arithmetic operations on complex numbers
    - Solved problem on taking square root of complex number
    - Miscellaneous problems on complex numbers
    - Advanced problem on complex numbers
    - Solved problems on de'Moivre formula
    - A curious example of an equation in complex numbers which HAS NO a solution
in this site.

The most relevant lessons marked (*) in the list.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic  "Complex numbers".


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Free of charge online textbook in ALGEBRA-II
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Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Put the numbers in format and use deMoivre's Theorem.

deMoivre's Theorem says that to find the n-th root of a complex number, you take the n-th root of the modulus and divide the angle by n.



For your problem, we are to find the values of . We have



The three cube roots of 8 are
(1)
(2)
(3)

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