# How do you use DeMoivre's theorem to simplify (sqrt3+i)^7?

Dec 21, 2016

The answer is $= 64 \left(- \sqrt{3} - i\right)$

#### Explanation:

De Moivre's theorem states

${\left(\cos \theta + i \sin \theta\right)}^{n} = \cos n \theta + i \sin n \theta$

Let $z = \sqrt{3} + i$

∥z∥=sqrt(3+1)=2

$z = 2 \cdot \left(\frac{\sqrt{3}}{2} + i \frac{1}{2}\right)$

$z = r \left(\cos \theta + i \sin \theta\right)$

$\cos \theta = \frac{\sqrt{3}}{2}$, $\implies$, $\theta = \frac{\pi}{6}$

$\sin \theta = \frac{1}{2}$, $\implies$, $\theta = \frac{\pi}{6}$

$z = 2 \left(\cos \left(\frac{\pi}{6}\right) + i \sin \left(\frac{\pi}{6}\right)\right)$

Therefore,

${z}^{7} = {2}^{7} {\left(\cos \left(\frac{\pi}{6}\right) + i \sin \left(\frac{\pi}{6}\right)\right)}^{7}$

$= 128 \left(\cos \left(\frac{7}{6} \pi\right) + i \sin \left(\frac{7}{6} \pi\right)\right)$

$= 128 \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)$

$= 64 \left(- \sqrt{3} - i\right)$

Dec 21, 2016

See explanation.

#### Explanation:

De Moivre's Theorem says that:

If a complex number $z$ is given in trigonometric form:

$z = r \left(\cos \varphi + i \sin \varphi\right)$

Then $n - t h$ power of $z$ is given as:

${z}^{n} = | z {|}^{n} \cdot \left(\cos n \varphi + i \sin n \varphi\right)$

So first thing to do is to change $z = \sqrt{3} + i$ into trigonometric form:

$| z | = \sqrt{{\sqrt{3}}^{2} + {1}^{2}} = \sqrt{3 + 1} = \sqrt{4} = 2$

$\cos \varphi = \frac{r e \left(z\right)}{r} = \frac{\sqrt{3}}{2} \implies \varphi = {30}^{o}$

So the trigonometric form of $z$ is:

$z = 2 \left(\cos 30 + i \sin 30\right)$

Now we can calculate ${z}^{7}$ according to the de Moivre's theorem:

${z}^{7} = {2}^{7} \cdot \left(\cos 7 \cdot 30 + i \sin 7 \cdot 30\right)$

${z}^{7} = 128 \cdot \left(\cos 210 + i \sin 210\right)$

${z}^{7} = 128 \cdot \left(\cos \left(180 + 30\right) + i \sin \left(180 + 30\right)\right)$

${z}^{7} = 128 \cdot \left(- \cos 30 - \sin 30 i\right)$

${z}^{7} = 128 \cdot \left(- \frac{\sqrt{3}}{2} - \frac{1}{2} i\right)$

Answer: ${z}^{7} = - 64 \sqrt{3} - 64 i$