Question

ofer coroana.
sa se calculeze

Answer 1

   
[tex]\displaystyle\\ \left|\left( \frac{ \sqrt{3} }{2} - \frac{1}{2}i \right)^{2016}\right| = ~?\\\\\\ \text{Modulul unui numar complex } z = a + bi \text{ este: }~|z|= \sqrt{a^2+b^2} \\\\ \text{Calculam modulul numarului complex dat in enunt:}\\\\ \left|\frac{ \sqrt{3} }{2} - \frac{1}{2}i \right| = \sqrt{\left(\frac{\sqrt{3} }{2}\right)^2 + \left(- \frac{1}{2}\right)^2}= \sqrt{\frac{3 }{4} + \frac{1}{4}}=\sqrt{\frac{4 }{4}}=\boxed{\bf 1}\\\\ \text{Folosim formula: }~~~ \boxed{\bf |z^n| = |z|^n } [/tex]


[tex]\displaystyle\\ \Longrightarrow~~~ \left|\left( \frac{ \sqrt{3} }{2} - \frac{1}{2}i \right)^{2016}\right| =\left| \frac{ \sqrt{3} }{2} - \frac{1}{2}i \right|^{2016} = 1^{2016} = \boxed{\bf 1}[/tex]