Question

(FASP) O valor de
${( \frac{ \sqrt{3} }{2} + \frac{1i}{2} )}^{10}$
é:

a)
$\frac{1}{2} - \frac{ \sqrt{3} }{2} \times i$
b)
$1 + i$
c)
$\frac{ \sqrt{3} }{2} + \frac{1i}{2}$
d)
$\frac{ - 1}{2} + \frac{ \sqrt{3} }{2} \times i$



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Answer 1

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$\sf z=\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}i\\\sf \rho=\sqrt{\bigg(\dfrac{\sqrt{3}}{2}\bigg)^2+\bigg(\dfrac{1}{2}\bigg)^2}\\\sf\rho=\sqrt{\dfrac{3}{4}+\dfrac{1}{4}}=\sqrt{1}=1\\\sf cos(\theta)=\dfrac{\frac{\sqrt{3}}{2}}{1}=\dfrac{\sqrt{3}}{2}\\\sf sen(\theta)=\dfrac{\frac{1}{2}}{1}=\frac{1}{2}\\\sf \theta=\dfrac{\pi}{6}\\\sf z=1\bigg[cos\bigg(\dfrac{\pi}{6}\bigg)+i~sen\bigg(\dfrac{\pi}{6}\bigg)\bigg]$

$\sf z^{10}=1^{10}\cdot\bigg[cos\bigg(10\cdot\dfrac{\pi}{6}\bigg)+i~sen\bigg(10\cdot\dfrac{\pi}{6}\bigg) \bigg]\\\sf z^{10}=cos\bigg(\dfrac{5\pi}{3}\bigg)+i~sen\bigg(\dfrac{5\pi}{3}\bigg)\\\large\boxed{\boxed{\boxed{\boxed{\sf\bigg(\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}i\bigg)^{10}=\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i}}}}$