How to use DeMoivre's Theorem to find (1+i)20 in standard form?

Question

How to use DeMoivre&#039;s Theorem to find (1+i)20 in standard form?

quenjvt · Accepted Answer

The formula for De Moivre, which is derived from Euler's formula that,

e^{i θ} = \cos (θ) + i \sin (θ)

, states that

{(\cos (θ) + i \sin (θ))}^{n} = \cos (n θ) + i \sin (n θ)

We are unable to use the formula directly because 1+i does not lie on the unit circle. To fix this, we can divide 1+i by its norm

\sqrt{2}

and factor out the necessary constant:

{(1 + i)}^{20} = {(\sqrt{2} (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i))}^{20}

= {(\sqrt{2})}^{20} {(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i)}^{20}

= 1024 {(\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))}^{20}

= 1024 (\cos (20 \cdot \frac{π}{4}) + i \sin (20 \cdot \frac{π}{4}))

= 1024 (\cos (5 π) + i \sin (5 π))

= 1024 (- 1 + i \cdot 0)

= - 1024

How to use DeMoivre's Theorem to find (1+i)^(20) in standard form?