Question

Help please will medal
The complex fourth roots of 5-5sqrt3i .

Answer 1

Convert your complex number into polar form then fourth root it!

Answer 2

@Bobo-i-bo would it be 0?

Answer 3

so you're saying that:
\[\sqrt[4]{5-5\sqrt{3}i}=0?\]

Answer 4

Or in other words:
\[0^4=5-5\sqrt3 i~ ?\]

Answer 5

No corverted it to trig form and got cos(pi/4) and sin(-pi/4) then multiplyed by 5sqrt2

Answer 6

This may help: http://tutorial.math.lamar.edu/Extras/ComplexPrimer/Roots.aspx :P

Answer 7

you can't really be expected to get this unless you push the number into polar form and know how to apply complex polar exponents...... called the DeMoivre theorem/whatever

so:

\(5-5\sqrt3i = 5 (1-\sqrt3i) = 5 (2)(\frac{1}{2} -\frac{\sqrt3}{2}i)\)

\(= 10 (\cos \frac{\pi}{3} - i \sin \frac{\pi}{3}) \)

[\(= 10 (\cos -\frac{\pi}{3} + i \sin -\frac{\pi}{3}) \)]

and you have \(10 e^{-\frac{\pi}{3} }\)

so you want \((10 e^{-\frac{\pi}{3} })^{1/4}\)

but then you have to factor in that you are spinning in a circle. so you really have \((10 e^{-\frac{\pi}{3} + 2n \pi })^{1/4}\)

so you get a bunch of roots but they settle into a pattern.