Convert the numbers to polar form.
#color(red)(z_1=4-5i)#
#rArrr_1=sqrt(4^2+(-5)^2)=sqrt(16+25)=sqrt41#
#rArrtheta_1=arctan((-5)/4)~~5.387#
#rArrr_1[costheta_1+isintheta_1]=color(red)(sqrt41[cos(5.387)+isin(5.387)])#
#color(blue)(z_2=2+7i)#
#rArrr_2=sqrt(2^2+7^2)=sqrt(4+49)=sqrt53#
#rArrtheta_2=arctan(7/2)~~1.292#
#rArrr_2[costheta_2+isintheta_2]=color(blue)(sqrt53[cos(1.292)+isin(1.292)])#
Now to multiply them together, the result will be:
#z_1z_2=r_1r_2[cos(theta_1+theta_2)+isin(theta_1+theta_2)]#
#rArrr_1r_2=sqrt41sqrt53=sqrt(41*53)=sqrt(2173)#
#rArrtheta_1+theta_2=arctan((-5)/4)+arctan(7/2)~~6.680#
We usually try to express #theta# on the interval
#0 < theta < 2pi#
#6.680 - 2pi~~0.396#
So our final answer is:
#color(purple)(z_1z_2=sqrt2173[cos(0.396)+isin(0.396)])#
