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

(iii) $-15-36i$

Find the square root of the following complex numbers:

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(iii) −15−36i-15-36i

Let z=−15−36iz=-15-36i.

∣z∣=(−15)2+(−36)2=1521=39|z|=\sqrt{(-15)^2+(-36)^2}=\sqrt{1521}=39

Here x=−15x=-15, y=−36lt0y=-36\\lt 0 so sgn⁡(y)=−1\operatorname{sgn}(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=±(23−33 i)\boxed{\sqrt{-15-36i}=\pm\left(2\sqrt{3}-3\sqrt{3}\,i\right)}

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