\begin{array}{lll} & 2)a) & \sqrt[3]{j}\end{array} r=\sqrt{0+1} r=1 \theta=\tan ^{-1}\left(\frac{1}{0}\right)=\tan ^{-1}(\infty) \theta=\frac{\pi}{2} \Rightarrow \quad z=\cos \left(\frac{\pi}{2}+2 k \pi\right)+j \sin \left(\frac{\pi}{2}+2 k \pi\right) \Rightarrow\quad z^{1/3}=\mleft[\cos \mleft(\frac{\pi}{2}+2k\pi\mright)+j\sin \mleft(\frac{\pi}{2}+2k\pi\mright)\mright]^{1/3} Here - z= 0+1.j \Rightarrow

\quad z^{1 / 3}=\left[\cos \left(\frac{\frac{\pi}{2}+2 k \pi}{3}\right)+j \sin \left(\frac{\frac{\pi}{2}+2 k \pi}{3}\right)\right] put k = 0 \Rightarrow z^{1 / 3}=\left[\cos \frac{\pi}{6}+j \sin \left(\frac{\pi}{6}\right)\right]=\frac{1}{2}(\sqrt{3}+j) k = 1 \Rightarrow z^{1 / 3}=\left[\cos \left(\frac{5 \pi}{6}\right)+j \sin \left(\frac{5 \pi}{6}\right)\right]=-\frac{1}{2}(\sqrt{3}-i) k=2 \Rightarrow \quad z^{1 / 3}=\left[\cos \left(\frac{9 \pi}{6}\right)+j \sin \left(\frac{9 \pi}{6}\right)\right]=\cos \left(\frac{3 \pi}{2}\right)+j \sin \left(\frac{3 \pi}{2}\right) \Rightarrow \quad z^{1 / 3}=-j \text { 10; } \sqrt[3]{j}=\frac{1}{2}(\sqrt{3}+j),-\frac{1}{2}(\sqrt{3}-j) ;-j \text { (2) (b) } \sqrt{2+2 j \sqrt{3}} \text { Hure } z=2+2 \sqrt{3} j \gamma=\sqrt{4+12} \gamma=\sqrt{16} T:4 \theta=\tan ^{-1}\left(\frac{2 \sqrt{3}}{2}\right)=\tan ^{-1}(\sqrt{3}) \theta=\frac{\pi}{3} \Rightarrow \quad z=4\left(\cos \left(\frac{\pi}{3}+2 k \pi\right)+j \sin \left(\frac{\pi}{3}+2 k \pi\right)\right) \Rightarrow \quad z^{1 / 2}=2\left[\cos \left(\frac{\pi}{3}+2 k \pi\right)+j \sin \left(\frac{\pi}{3}+2 k \pi\right)\right]^{1 / 2} \Rightarrow \quad z^{1 / 2}=2\left\{\cos \left(\frac{\frac{\pi}{3}+2 k \pi}{2}\right)+j \sin \left(\frac{\frac{\pi}{3}+2 k \pi}{2}\right)\right\} put k=0 \Rightarrow \quad z^{1 / 2}=2\left(\cos \left(\frac{\pi}{6}\right)+j \sin \left(\frac{\pi}{6}\right)\right) =2\left(\frac{\sqrt{3}}{2}+j \cdot \frac{1}{2}\right) \Rightarrow 2 \times \frac{1}{2}(\sqrt{3}+j)=\sqrt{3}+j put k= 1 \Rightarrow \quad z^{1 / 2}=2\left(\cos \left(\pi+\frac{\pi}{6}\right)+j \sin \left(\pi+\frac{\pi}{6}\right)\right) =\left(-\cos \left(\frac{\pi}{6}\right)-j \sin \left(\frac{\pi}{6}\right)\right) \times 2 =-\frac{1}{2}(\sqrt{3}+j) \times 2=-(\sqrt{3}+j) so,z^{1/2}=(\sqrt{3}+j),-(\sqrt{3}+j) \text { (2) } \text { Hent } \quad z=-\frac{1}{2}-\frac{\sqrt{3}}{2} j \gamma=\sqrt{\frac{1}{4}+\frac{3}{4}} \left[\begin{array}{l} \gamma=1 \\ \hline \end{array}\right. \arg (z)=\theta=-\pi+\tan ^{-1}\left(\frac{\sqrt{3}}{1}\right) \theta=-\pi+\frac{\pi}{3} \theta=\frac{-2\pi}{3} z=\cos \left(\frac{-2 \pi}{3}+2 k \pi\right)+j \sin \left(\frac{-2 \pi}{3}+2 k \pi\right) z^{1 / 3}=\cos \left(\frac{-\frac{2 \pi}{3}+2 k \pi}{3}\right)+j \sin \left(\frac{-2 \pi+2 k \pi}{3}\right) z^{1 / 3}=\cos \left(\frac{-2 \pi}{9}\right)+j \sin \left(\frac{-2 \pi}{9}\right) z^{1 / 3}=\cos \left(\frac{2 \pi}{9}\right)-j \sin \left(\frac{2 \pi}{9}\right) z^{1 / 3}=\operatorname{cod}\left(\frac{4 \pi}{9}\right)+j \sin \left(\frac{4 \pi}{9}\right) \Rightarrow \quad z^{1 / 3}=\cos \left(\frac{10 \pi}{9}\right)+j \sin \left(\frac{10 \pi}{9}\right) ^^20so;^^20\quad \sqrt[3]{\frac{-1-j \sqrt{3}}{2}}=\cos \mleft(\frac{2 \pi}{9}\mright)-j\sin \mleft(\frac{2 \pi}{9}\mright) \cos \left(\frac{4 \pi}{9}\right)+j \sin \left(\frac{4 \pi}{9}\right) \cos \left(\frac{10 \pi}{9}\right)+j \sin \left(\frac{10 \pi}{9}\right) put k=2 put k = 1 put k=o but k=2 ^^20(2)^^20c)\quad \sqrt[3]{\frac{-1-j \sqrt{3}}{2}}