Question

How do you find the 3rd root of `8e^(45i)`?

Answer 1

Roots are `2[cos (pi/12) +isin(pi/12)], 2[cos ((3pi)/4) +((isin(3pi))/4)] & 2[cos ((17pi)/12) +((isin(17pi))/12)]`

Explanation:

`8e^(45i)=8(cos45+isin45)=8(cos (pi/4)+isin(pi/4)):.` 1st root: `[8(cos pi/4)+(isinpi/4)]^(1/3)= 8^(1/3)[ cos (pi/(4*3)) +isin(pi/(4*3)]=2[cos (pi/12) +isin(pi/12)] :.`for 2nd root `(2pi)/3` to be added to the angle i.e `theta=(pi/12+(2pi)/3)=(9pi)/12=(3pi)/4:.`2nd root is `2[cos ((3pi)/4) +((isin(3pi))/4)]`
for 3rd root `(2pi)/3` to be added to the angle i.e `theta=((3pi)/4+(2pi)/3)=(17pi)/12:.`3rd root is `2[cos ((17pi)/12) +((isin(17pi))/12)]`[Ans]