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

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

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Answer 1 · 2015-09-02T22:33:56

If you express your complex number in polar form as $r (cos θ + i sin θ)$ $r(cos theta + i sin theta)$ , then it has fourth roots:

$α = \sqrt[4]{r} (cos (\frac{θ}{4}) + i sin (\frac{θ}{4}))$ $alpha = root(4)(r)(cos (theta/4) + i sin (theta/4))$ , $i α$ $i alpha$ , $- α$ $-alpha$ and $- i α$ $- i alpha$

Explanation:

Given $a + i b$ $a+ib$ , let $r = \sqrt{a^{2} + b^{2}}$ $r = sqrt(a^2+b^2)$ , $θ = atan2 (b, a)$ $theta = "atan2"(b, a)$

Then $a + i b = r (cos θ + i sin θ)$ $a + ib = r (cos theta + i sin theta)$

This has one $4 t h$ $4th$ root $α = \sqrt[4]{r} (cos (\frac{θ}{4}) + i sin (\frac{θ}{4}))$ $alpha = root(4)(r)(cos (theta/4) + i sin (theta/4))$

There are three other $4 t h$ $4th$ roots: $i α$ $i alpha$ , $- α$ $-alpha$ and $- i α$ $-i alpha$

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

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