How do you use DeMoivre's theorem to simplify ${(1 + i)}^{4}$ $(1+i)^4$ ?

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How do you use DeMoivre's theorem to simplify ${(1 + i)}^{4}$ $(1+i)^4$ ?

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Answer 1 · 2016-09-11T23:12:25

${(1 + i)}^{4} = - 4$ $(1+i)^4 = -4$

Explanation:

Note that

$1 + i = \sqrt{2} (cos (\frac{π}{4}) + i sin (\frac{π}{4}))$ $1+i = sqrt(2)(cos (pi/4) + i sin (pi/4))$

De Moivre tells us that:

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

So we find:

${(1 + i)}^{4} = {(\sqrt{2} (cos (\frac{π}{4}) + i sin (\frac{π}{4})))}^{4}$ $(1+i)^4 = (sqrt(2)(cos (pi/4) + i sin (pi/4)))^4$

${(1 + i)}^{4} = {(\sqrt{2})}^{4} {(cos (\frac{π}{4}) + i sin (\frac{π}{4}))}^{4}$ $color(white)((1+i)^4) = (sqrt(2))^4(cos (pi/4) + i sin (pi/4))^4$

${(1 + i)}^{4} = 4 (cos π + i sin π)$ $color(white)((1+i)^4) = 4(cos pi + i sin pi)$

${(1 + i)}^{4} = 4 ((- 1) + i (0))$ $color(white)((1+i)^4) = 4((-1) + i (0))$

${(1 + i)}^{4} = - 4$ $color(white)((1+i)^4) = -4$

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How do you use DeMoivre's theorem to simplify ${(1 + i)}^{4}$ $(1+i)^4$ ?

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