diytechowto
diytechowto diytechowto
  • 17-04-2019
  • Mathematics
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Compute the indicated power by using DeMoivre’s theorem:

Compute the indicated power by using DeMoivres theorem class=

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LammettHash
LammettHash LammettHash
  • 17-04-2019

[tex]2+2i=2\sqrt2e^{i\pi/4}[/tex]

[tex]3-3i=3\sqrt2e^{-i\pi/4}[/tex]

[tex]\implies\dfrac{2+2i}{3-3i}=\dfrac23e^{i\pi/2}[/tex]

By DeMoivre's theorem,

[tex]\left(\dfrac{2+2i}{3-3i}\right)^5=\left(\dfrac23\right)^5e^{i5\pi/2}=\dfrac{32}{243}e^{i\pi/2}=\dfrac{32i}{243}[/tex]

Just to confirm:

[tex]\dfrac{2+2i}{3-3i}\cdot\dfrac{3+3i}{3+3i}=\dfrac{12i}{18}=\dfrac{2i}3\implies\left(\dfrac{2+2i}{3-3i}\right)^5=\dfrac{32i}{243}[/tex]

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