How do I find the fifth root of a complex number?

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How do I find the fifth root of a complex number?

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Answer 1 · 2015-11-01T09:32:16

Convert to polar form first, then...

Explanation:

A Complex number in the form $r (cos θ + i sin θ)$ $r(cos theta + i sin theta)$ has $5$ $5$ th roots:

$\sqrt[5]{r} (cos (\frac{θ}{5}) + i sin (\frac{θ}{5}))$ $root(5)(r)(cos (theta/5) + i sin (theta/5))$

$\sqrt[5]{r} (cos (\frac{θ + 2 π}{5}) + i sin (\frac{θ + 2 π}{5}))$ $root(5)(r)(cos ((theta + 2pi)/5) + i sin ((theta+2pi)/5))$

$\sqrt[5]{r} (cos (\frac{θ + 4 π}{5}) + i sin (\frac{θ + 4 π}{5}))$ $root(5)(r)(cos ((theta + 4pi)/5) + i sin ((theta+4pi)/5))$

$\sqrt[5]{r} (cos (\frac{θ + 6 π}{5}) + i sin (\frac{θ + 6 π}{5}))$ $root(5)(r)(cos ((theta + 6pi)/5) + i sin ((theta+6pi)/5))$

$\sqrt[5]{r} (cos (\frac{θ + 8 π}{5}) + i sin (\frac{θ + 8 π}{5}))$ $root(5)(r)(cos ((theta + 8pi)/5) + i sin ((theta+8pi)/5))$

Conventionally your original $θ$ $theta$ is in the range $(- π, π]$ $(-pi, pi]$ or the range $[0, 2 π)$ $[0, 2pi)$ according to your definition of $A r g (z)$ $Arg(z)$ and the first of these five roots is the Principal Complex fifth root.

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How do I find the fifth root of a complex number?

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