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Find the square roots of: $8-15i.$

Could I get some working out to solve it?

Also what are different methods of doing it?

amWhy
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Alex
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    Take a look at trigonometrical form of a complex number and de Moivre's formula. – Andrei Rykhalski Oct 12 '14 at 12:48
  • See this. – Git Gud Oct 12 '14 at 12:48
  • Very suggestively have $|8-15i|=\sqrt{8^2+15^2}=17$. No guarantees, but this suggests that we might try numbers with absolute value $\sqrt{17}$. The simplest of those are $4\pm i$. We have $$(4+i)^2=16+8i+i^2=15+8i=i(8-15i).$$ Damn! So close! How could we cancel that extra $i$? Do you know how to find the square roots of $i$ with Moivre's formula? – Jyrki Lahtonen Oct 12 '14 at 13:37

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You might start with solving $$ (x + i y)^2 = 8 - 15 i $$

component-wise (real part vs imaginary part), which leads to two equations in the two unknowns $x$ and $y$.

More general is using Euler's formula $e^{i \phi} = \cos \phi + i \sin \phi$, using polar coordinates $(r, \phi)$, $z = r e^{i \phi}$ with $z = x + i y$, $r = |z| = \sqrt{x^2 + y^2}$ and $\tan \phi = y / x$.

mvw
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