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

(ii) $8-6i$

Find the square root of the following complex numbers:

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(ii) 8−6i8-6i

Let z=8−6iz=8-6i.

∣z∣=82+(−6)2=100=10|z|=\sqrt{8^2+(-6)^2}=\sqrt{100}=10

Here x=8x=8, y=−6lt0y=-6\\lt 0 so sgn⁡(y)=−1\operatorname{sgn}(y)=-1.

8−6i=±(10+82−i10−82)=±(9−i1)=±(3−i)\begin{aligned} \sqrt{8-6i} &=\pm\left(\sqrt{\frac{10+8}{2}}-i\sqrt{\frac{10-8}{2}}\right)\\ &=\pm\left(\sqrt{9}-i\sqrt{1}\right)\\ &=\pm(3-i) \end{aligned} 8−6i=±(3−i)\boxed{\sqrt{8-6i}=\pm(3-i)}

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