(iii) −15−36i-15-36i−15−36i Let z=−15−36iz=-15-36iz=−15−36i. ∣z∣=(−15)2+(−36)2=1521=39|z|=\sqrt{(-15)^2+(-36)^2}=\sqrt{1521}=39∣z∣=(−15)2+(−36)2=1521=39 Here x=−15x=-15x=−15, y=−36lt0y=-36\\lt 0y=−36lt0 so sgn(y)=−1\operatorname{sgn}(y)=-1sgn(y)=−1. −15−36i=±(39−152−i39+152)=±(12−i27)=±(23−33 i)\begin{aligned} \sqrt{-15-36i} &=\pm\left(\sqrt{\frac{39-15}{2}}-i\sqrt{\frac{39+15}{2}}\right)\\ &=\pm\left(\sqrt{12}-i\sqrt{27}\right)\\ &=\pm\left(2\sqrt{3}-3\sqrt{3}\,i\right) \end{aligned}−15−36i=±(239−15−i239+15)=±(12−i27)=±(23−33i) −15−36i=±(23−33 i)\boxed{\sqrt{-15-36i}=\pm\left(2\sqrt{3}-3\sqrt{3}\,i\right)}−15−36i=±(23−33i)