Complex variables

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Complex variables

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if alpha and beta are imaginary cube roots of unity then find the value of Alpha power 4 + beta power 4 + alpha beta

ARYAN SAGAR 11th Jan, 2021

Dear student,

The cube roots of unity are 1, w, w^2.
And according to the property of cube roots of unity :- 1 + w + w^2 = 0

Now, if alpha and beta are imaginary cube roots of unity, so take alpha as w and beta as w^2 ( since 1 is a real root ).

Now, (alpha)^4 + (beta)^4 +(alpha beta) = w^4 + (w^2)^4 + (ww^2).

Now, we get :-
w^4 + w^6 + w^3

Taking w^3 as common we get :-

w^3 ( 1 + w + w^2 )

And 1 + w + w^2 = 0.

So, the value of above equation is 0.
Therefore answer is = 0.

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find valu of x and y . x+y iota upon 2+3iota + 2+iota upon 2-3iota = 9 upon13 (1+iota)

Anuj More 19th Oct, 2020

Dear Student,

You can find the values of x & y by multiplying & dividing (2-3i) to the first term (i.e by its complex conjugate) & multiply & divide second term by (2+3i) (complex conjugate of 2-3i). Similarly solve the equations on LHS till you get an equation in terms of A+Bi. Similarly multiply 1-i to RHS i.e by its complex conjugate. Equate the real & imaginary parts on both sides. You will find 2 equations in terms of x & y. Solve them simultaneously and you will find X & y in fractions.

Hope I was able to solve your query.

Hope this helps.

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