If `Z^5` is a non-real complex number, then find the minimum value of `(Imz^5)/(Im^5z)`

Answer

`-4`

Explanation

Let z= a+ ib,
`b ne 0` where Im Z = b
`therefore " " Z^(5) = (a+ib)^(5)`
`= a^(5) + 5a^(4) bi + 10^(3)b^(2)i^(2) + 10a^(2)b^(3)i^(2) + 5ab^(4)i^(4) + i^(5)b^(5)`
`therefore" " Lm z^(5)=5a^(4)b- 10a^(2)b^(3) + b^(5)`
`y= (ImZ^(5))/((ImZ^(5)))=((a)/(5))^(2)-10((a)/(b))^(2)+1`
`= 5[((a)/(b))^(4)-2((a)/(b))^(2)+(1)/(5)]`

`=5[(((a)/(b))^(2)-1)^(2) +(1)/(5)-1]`
`=5(((a)/(b))^(2)-1)^(2)=4`
So, `y_("min")= -4`