Theory of Intelligence
11TH•Mathematics•Complex Numbers•Question 7

(i) $-7-24i$

Find the square root of the following complex numbers:

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(i) −7−24i-7-24i

Let z=−7−24iz=-7-24i.

∣z∣=(−7)2+(−24)2=625=25|z|=\sqrt{(-7)^2+(-24)^2}=\sqrt{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)

Here x=−7x=-7, y=−24<0y=-24<0 so sgn⁡(y)=−1\operatorname{sgn}(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=±(3−4i)\boxed{\sqrt{-7-24i}=\pm(3-4i)}

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