(i) −7−24i-7-24i−7−24i Let z=−7−24iz=-7-24iz=−7−24i. ∣z∣=(−7)2+(−24)2=625=25|z|=\sqrt{(-7)^2+(-24)^2}=\sqrt{625}=25∣z∣=(−7)2+(−24)2=625=25 Use the standard square-root form: x+iy=±(∣z∣+x2+i sgn(y)∣z∣−x2)\sqrt{x+iy}=\pm\left(\sqrt{\frac{|z|+x}{2}}+i\,\operatorname{sgn}(y)\sqrt{\frac{|z|-x}{2}}\right)x+iy=±(2∣z∣+x+isgn(y)2∣z∣−x) Here x=−7x=-7x=−7, y=−24<0y=-24<0y=−24<0 so sgn(y)=−1\operatorname{sgn}(y)=-1sgn(y)=−1. −7−24i=±(25−72−i25+72)=±(9−i16)=±(3−4i)\begin{aligned} \sqrt{-7-24i} &=\pm\left(\sqrt{\frac{25-7}{2}}-i\sqrt{\frac{25+7}{2}}\right)\\ &=\pm\left(\sqrt{9}-i\sqrt{16}\right)\\ &=\pm(3-4i) \end{aligned}−7−24i=±(225−7−i225+7)=±(9−i16)=±(3−4i) −7−24i=±(3−4i)\boxed{\sqrt{-7-24i}=\pm(3-4i)}−7−24i=±(3−4i)