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

(iv) $119+120i$

Find the square root of the following complex numbers:

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(iv) 119+120i119+120i

Let z=119+120iz=119+120i.

∣z∣=1192+1202=28561=169|z|=\sqrt{119^2+120^2}=\sqrt{28561}=169

Here x=119x=119, y=120>0y=120>0 so sgn⁡(y)=+1\operatorname{sgn}(y)=+1.

119+120i=±(169+1192+i169−1192)=±(144+i25)=±(12+5i)\begin{aligned} \sqrt{119+120i} &=\pm\left(\sqrt{\frac{169+119}{2}}+i\sqrt{\frac{169-119}{2}}\right)\\ &=\pm\left(\sqrt{144}+i\sqrt{25}\right)\\ &=\pm(12+5i) \end{aligned} 119+120i=±(12+5i)\boxed{\sqrt{119+120i}=\pm(12+5i)}

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