The exponential form of a complex number #z=a+ib# is
#z=re^(itheta)#
where,
#r=|z|=sqrt(a^2+b^2)#
#costheta=a/|z|#
#sintheta=b/|z|#
The complex number is
#z=z_1/z_2=(6-6i)/(-sqrt3+i)#
For the numerator,
#z_1=6-6i=6sqrt2(1/sqrt2-1/sqrt2i)#
#costheta_1=1/sqrt2#
#sintheta_1=-1/sqrt2#
#theta_1=-1/4pi# #[mod 2pi]#
#z_1=6sqrt2e^(-pi/4i)#
For the denominator,
#z_2=-sqrt3+i=2(-sqrt3/2+1/2i)#
#costheta_2=-sqrt3/2#
#sintheta_2=1/2#
#theta_2=5/6pi#, #[mod 2pi]#
#z_=2e^(5/6pii)#
Finally,
#z=z_1/z_2=(6sqrt2e^(-pi/4i))/(2e^(5/6pii))#
#=3sqrt2e^((-pi/4-5/6pi)i)#
#=3sqrt2e^(-13/12pii)#
#=3sqrt2e^(11/12pii)#, #[mod 2pi]#
