Question

Complex numbers....

Answer 1

If z^3 = 8*i, give one root of the equation in exact polar form, and another INEQUIVALENT root. How many inequivalent roots are there?

Answer 2

i guess there are 3
do you know how to start?

Answer 3

first write
\[8i\] in polar (trigonometric) form as
\[8i=8(\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}))\] then you want
\[\left(8(\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}))\right)^{\frac{1}{3}} \]

Answer 4

take the cube root of 8, which is 2, and divide the angle by 3, so you get
\[29\cos(\frac{\pi}{6})+i\sin(\frac{\pi}{6}))\] then evaluate sine and cosine of those numbers

Answer 5

typo there
\[2(\cos(\frac{\pi}{6})+i\sin(\frac{\pi}{6}))\]

Answer 6

you get
\[2(\frac{\sqrt{3}}{2}+i\frac{1}{2})\]
\[\sqrt{3}+i\] is one solution

Answer 7

Hey thanks heaps for this - sorry I didn't reply.. It was quite late in my time zone :) cheers!