Answer:
a=64 and b=0
Step-by-step explanation:
[tex]z=-1-\sqrt{3}i[/tex]
use [tex]z=e^{ix}=cosx+isinx[/tex]
Now we find out x using given z
[tex]z=-1-\sqrt{3}i[/tex]
Multiply and divide the right side by -2
[tex]z=-2(\frac{1}{2} +\frac{\sqrt{3}}{2})[/tex]
We know cos(pi/3) = 1/2
sin(pi/3)= sqrt(3)/2
[tex]z=-2(\frac{1}{2} +\frac{\sqrt{3}}{2})[/tex]
[tex]z=-2(cos(\frac{\pi}{3})+sin(\frac{\pi}{3})[/tex]
compare the above equation with use [tex]z=e^{ix}=cosx+isinx[/tex]
x= pi/3
so [tex]z=-2e^{\frac{\pi}{3}i}[/tex]
[tex]z^6=64(e^{\frac{\pi}{3}i})^6[/tex]
[tex]z^6=64e^{\frac{6\pi}{3}i}[/tex]
[tex]z^6=64e^{2\pi*i}[/tex]
Plug in 2pi in the z equation
[tex]z=64e^{2\pi*i}=64(cos(2\pi)+isin(2\pi))[/tex]
So z= 64(1 + i(0))
z=64
So a= 64 and b=0
