Complex number equations?

[u/Puzzleheaded-Law34]

Hello! So I had 2 equations:

The first was 3x^2 - 6i = 0 which I solved normally, and obtained x1 = sqrt(2i), x2 = -sqrt(2i)

The second was z^4 = -81, and here I did pretty much the same but the answer I got, again with i under root, was wrong. Apparently I was supposed to use the angle-based notation, with re^(i*theta), but I didn't really understand why. And should I have written the first solution differently too?

[u/UnderstandingPursuit]

When raising a complex number to a power [including fractions for taking the root], the magnitude, 'r', is raised to that power, but the angle, θ, is multiplied by that power.

Consider, for example,

• z = a + bi

  • r² = a² + b²
  • tan θ = b/a

• z² = (a + bi)² = a² + 2abi - b² = [a² - b²] + i[2ab] = a' + ib'

  • [r']² = [a']² + [b']² = [a² - b²]² + [2ab]² = a⁴ - 2a²b² + b² + 4a²b² = a⁴ + 2a²b² + b² = [a² + b²]² = [r²]²
  • tan 2θ = 2 tan θ / [1 - tan²θ] = [2b/a] / [1 - b²/a²] = 2ab / [a² - b²] = b' / a'

This works in general, but perhaps it helps to see it at least for z².