$$z\ =\sqrt[]{2\ }\left(\cos45\degree+i\ \sin45\degree\right)$$

$$\frac{3}{2}x+2y$$

$$z=\sqrt[]{2}e^{i\ 45\degree}$$

$$z=3+2i$$

$$4+3i$$

$$3+2i$$

$$1+i$$

$$3-2i$$

$$6+33i$$

$$5-33i$$

$$6-33i$$

$$5+33i$$

$$9-\sqrt[]{2}i$$

$$8+\sqrt[]{2}i$$

$$9+\sqrt[]{2}i$$

$$8-\sqrt[]{2}i$$

$$\frac{5}{28}-\frac{\sqrt[]{3}}{28}i$$

$$\frac{5}{29}+\frac{\sqrt[]{4}}{29}i$$

$$\frac{5}{28}+\frac{\sqrt[]{3}}{28}i$$

$$\frac{5}{29}-\frac{\sqrt[]{4}}{29}i$$

$$6+5i$$

$$\frac{6}{61}-\frac{5}{61}i$$

$$\frac{6}{61}+\frac{5}{61}i$$

$$6-5i$$

$$16\ +i$$

$$2+5i$$

$$2-5i$$

$$16-i$$

$$2\sqrt[]{3}+\sqrt[]{6}$$

$$\sqrt[]{6}$$

$$2\sqrt[]{3}$$

$$2\sqrt[]{3}-\sqrt[]{6}$$

$$Arg\left(z\right)=45\degree+2k\pi,\ k\ \epsilon\ Z$$

$$Arg\left(z\right)=45\degree-2k\pi,\ k\ \epsilon\ Z$$

$$Arg\left(z\right)=30\degree-2k\pi,\ k\ \epsilon\ Z$$

$$Arg\left(z\right)=30\degree+2k\pi,\ k\ \epsilon\ Z$$