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hello, how do i solve : (Z-2i)^4 = -8-8i sqrt3 thanks in advance

hello,
 
how do i solve : (Z-2i)^4 = -8-8i sqrt3
 
thanks in advance

Grade:12th pass

1 Answers

Aditya Gupta
2081 Points
5 years ago
this ques is EXTREMELY easy.
all we need to do is to write RHS in re^itheta form and then apply de moivre rule.theorem.
note that -8-8i sqrt3= 16*e^(i4π/3)
also 16=2^4. let z= 2x
so we have (x – i)^4 = e^(i4π/3)
now x – i= [e^(i4π/3)]1/4 , which will have four distinct values due to moivres rule.
hence x= i + [e^(i4π/3)]1/4 
or z= 2i+2[e^(i4π/3)]1/4 
note that it is fairly elementary and easy to apply moivres theorem on [e^(i4π/3)]1/4 so you can do that on your own

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