What is the easiest way to solve $z^3 = 1 + i$? I thought of remembering the unity roots of $$z^3, z^4, z^5,... z^n$$ but it sucks. I could use Euler's form or trigonometric but it's hard to get back to cartesian (rectangular) form. any ideas ? How would you solve any equations in the form $$z^n=c$$ $c$ is a complex number
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4Write $1+i$ as $re^{i\theta}$? – lulu Feb 05 '24 at 16:54
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ok and then, I know it's a really simple case, sorry for that. That's what I did and got $$e^{\frac{\pi*i}{4}}$$ – Barthélémy Coquoz Feb 05 '24 at 16:56
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Use De Moivre's Formula: If $z = r(\cos\theta + i\sin\theta)$, then $z^n = r^n(\cos(n\theta) + i\sin(n\theta))$. – Dan Feb 05 '24 at 16:57
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1It's often useful to know that $(1+i)^2 = 2i$. – MJD Feb 05 '24 at 17:07
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Possibly helpful: How to solve $x^3=-1$ – MJD Feb 05 '24 at 17:08
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Let $c=re^{i\phi}$. Then $$ c^{\frac1n}=r^{\frac1n}e^{i\left(\frac\phi n+\frac{2\pi}n k\right)} =r^{\frac1n}\left[\cos\left(\frac\phi n+\frac{2\pi}n k\right)+i\sin\left(\frac\phi n+\frac{2\pi}n k\right)\right], $$ with $k=0\dots n-1$.
In your example $r=\sqrt2$, $\phi=\frac\pi4$.
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