3Edemoivre


 * De Moivre's Theorem **

math \text {For } z = r \text{cis} \big( \theta \big) math

De Moivre's Theorem gives us the nth power of z math . \qquad z^n = r^n \text{cis} \big( n\theta \big) \quad n \in Q \qquad \{ \text {Rational numbers} \} math

Note Compare this with the index form equivalent math . \qquad z^n = \left( re^{\theta i} \right)^n = r^n e^{n\theta i} math

De Moivre's theorem is revealed as an application of basic index laws.


 * Example 1 **

math \\ \text{For } \; z = 3 \text {cis} \left( \dfrac {\pi}{3} \right) \; \text{ find } \; z^5 math


 * Solution:**

math \\ . \qquad z^5 = 3^5 \text {cis} \left( \dfrac {5\pi}{3} \right) \qquad \textit {angle outside domain so subtract } 2\pi \\. \\ . \qquad z^5 = 243 \text {cis} \left( \dfrac {-\pi}{3} \right) math


 * Example 2 **

math \text {For } z = 2 + 2i \; \text { find } \; z^4 math


 * Solution:**


 * The easiest way is to convert to Polar Form and use De Moivre'sTheorem

math \\ . \qquad r = \sqrt{2^2 + 2^2} \\. \\ . \qquad \quad = \sqrt {8} \\. \\ . \qquad \theta = Tan^{-1} \left( \dfrac{2}{2} \right) \\. \\ . \qquad \quad = Tan^{-1} (1) \\. \\ . \qquad \quad = \dfrac {\pi}{4} math

so we now have z in polar form: math . \qquad z = \sqrt{8} \text {cis} \left( \dfrac {\pi}{4} \right) math

Using De Moivre's Theorem math \\ . \qquad z^4 = \left( \sqrt{8} \right)^4 \text {cis} \left( \dfrac {4\pi}{4} \right) \\. \\ . \qquad z^4 = 64 \text {cis} \big( \pi \big) \\. \\ . \qquad z^4 = -64 + 0i \\. \\ . \qquad z^4 = -64 math

De Moivre's Theorem can be used to find the square roots, cube roots, etc of a complex number.

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