Question

((2<45°)^2+(3<120°))/((3<120°)+(4<180°)) leave your answer in rectangular form

Answer 1

$$(z_1)^2=(2)^2\angle(2\cdot45\degree)=4i$$

$$(z_1)^2+z_2=4i+3(-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i)$$

$$=-\dfrac{3}{2}+\dfrac{8+3\sqrt{3}}{2}i$$

$$z_2+z_3=3(-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i)-4=-\dfrac{11}{2}+\dfrac{3\sqrt{3}}{2}i$$

$$\dfrac{(z_1)^2+z_2}{z_2+z_3}=\dfrac{-3+(8+3\sqrt{3})i}{-11+3\sqrt{3}i}$$

$$=\dfrac{(-3+(8+3\sqrt{3})i)(-11-3\sqrt{3}i)}{121+27}$$

$$=\dfrac{33+9\sqrt{3}i-88i-33\sqrt{3}i+24\sqrt{3}+27}{148}$$

$$=\dfrac{15+6\sqrt{3}}{37}-\dfrac{22+6\sqrt{3}}{37}i$$