Find the fifth roots of $-3+3i$ in exponential form.
My answers are: $$1.335e^{3i\pi/20}$$ $$1.335e^{11i\pi/20}$$ $$1.335e^{19i\pi/20}$$ $$1.335e^{27i\pi/20}$$ $$1.335e^{35i\pi/20}$$
Wolfram gives the last two as $1.335e^{-13i\pi/20}$ and $1.335e^{-i\pi/4}$ (link to Wolfram Alpha)
I can see these are my values minus $2\pi$, but why is it written like this, i.e. in negative form?
On
The polar form of a complex number is not unique. A common convention to choose a unique angle of a complex number is to return the angle in the range $-\pi < \theta \le \pi$. This is called the principle argument of the number, rather than just an argument of the number. But because you can go forward or backward one rotation about the origin and end up in the same spot, you have that you can add any multiple of $2\pi$ and still represent the same coordinate.
Algebraically, we have that $e^{2k\pi i} = 1$, so $r e^{i\theta} = r e^{i\theta}e^{2k\pi i} = r e^{i\theta + 2k \pi i}$ for any integer $k$.
Wolfram Alpha is using the principal branch of the complex logarithm (Wikipedia link), which is defined to have complex part in $(-\pi,\pi]$; equivalently, it is using the principal value of the complex argument function (Wikipedia link).