Esercizio. 
Risolvere in 
la seguente equazione
Risolvere in la seguente equazione
![\[x^2-6i=0\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-380c2678db41f19ef71db13ca697e1fd_l3.png "Rendered by QuickLaTeX.com")
Soluzione
Avendo un numero complesso 
, le radici 
-esime di 
sono
![\[\sqrt[n]{z} = \sqrt[n]{\vert z \vert} \left(\cos \left(\frac{\alpha}{n} + \frac{2k\pi}{n} \right) + i \sin \left(\frac{\alpha}{n} + \frac{2k\pi}{n} \right)\right) \qquad k=0,1,\dots,n-1.\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-2618caf6709126bc6404289f6cca20db_l3.png "Rendered by QuickLaTeX.com")
Nel nostro caso vogliamo trovare
![\[x^2 = 6i \quad \Rightarrow \quad x = \sqrt{6i}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-d3a8f99565fad71166bd1d869a38de85_l3.png "Rendered by QuickLaTeX.com")
quindi
![\[\vert z \vert = \sqrt{0^2+6^2} = 6 \qquad \qquad\mbox{e} \qquad\qquad \alpha = \dfrac{\pi}{2}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-fb897b4481f6ef1ce86bb63b1d0c8365_l3.png "Rendered by QuickLaTeX.com")
pertanto
![\[\sqrt{6i} = \sqrt{6} \left(\cos \left(\frac{\pi}{4} + k\pi \right)+ i \sin \left(\frac{\pi}{4} + k\pi \right)\right) \qquad k=0,1\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-776c76598380aa45147fe00d719dd8f1_l3.png "Rendered by QuickLaTeX.com")
Sostituendo 
abbiamo
![\[\begin{aligned} & k =0 \quad \Rightarrow \quad \sqrt{6i} = \sqrt{6} \left(\cos \left(\frac{\pi}{4}\right)+ i \sin \left(\frac{\pi}{4} \right)\right) = \sqrt{6}\left(\dfrac{\sqrt{2}}{2} + i \dfrac{\sqrt{2}}{2} \right) = \sqrt{3} + i\sqrt{3}\\\\ & k =1 \quad \Rightarrow \quad \sqrt{6i} = \sqrt{6} \left(\cos \left(\frac{\pi}{4}+\pi\right)+ i \sin \left(\frac{\pi}{4} +\pi\right)\right) = \sqrt{6}\left(\dfrac{-\sqrt{2}}{2} - i \dfrac{\sqrt{2}}{2} \right) = -\sqrt{3} - i\sqrt{3}\\\\ \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-10600a99a372caa2d2a62843d66c7eb0_l3.png "Rendered by QuickLaTeX.com")
e concludiamo che
![\[\boxed{ x = \pm \sqrt{3} (1+i) }\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-871240c7b33d3b5cb231d9b6f0d92e72_l3.png "Rendered by QuickLaTeX.com")
Fonte: Matematica.verde 3 – Zanichelli