Thread: Complex numbers, don't understand last bit! :(

1. Complex numbers, don't understand last bit! :(

Hey guys i was working through this question and when i cheked my answers i realised there was an extra bit to do to get the final answer but i dont know wt im supposed to do:

Q) Express (-1 + sqrt3i) in modulus-argument form. Evaluate (-1 + sqrt3i)^8,
expressing your answer in a + ib form.

I got z= 2cis 2pi/ 3 for mod -arg form but when i evaluate it I can get to:

2^8 cis 16pi/3

but then the answers tell me tht the answer above is equal to 2^8 cis 4pi/3???

And that the final answer in a+ ib form is : -128 - 128sqrt3i .

What i want to know is how do i go from 2^8 cis 16pi/3 ==> -128 - 128 sqrt3i??

2. You need to realize that $\displaystyle \frac{16\pi}{3}=4\pi+\color{blue}\frac{4\pi}{3}$.

3. i figured that was the case but how am i supposed to work that out for something like:

2^10 cis 40pi/3

is there a certain method etc?

4. $\displaystyle \frac{40 \pi}{3} = \frac{ (6 \cdot 3) \cdot 2 \pi + 4 \pi}{3}$ $\displaystyle = 12 \pi + \frac{4 \pi }{3}$

Simply find out the largest multiple of $\displaystyle 2 \pi$ times the denominator, and "subtract" it from the numerator - in this example, 6 * 3, and in your previous example, 2*3.