Originally Posted by

**topsquark** You know, I'm very glad I'm going through this little book. I thought I knew everything about doing these, but I had never considered a problem like this one.

The problem is to put

$\displaystyle \left ( 1 - \frac{\sqrt{3 - i}}{2} \right ) ^{24}$

in rectangular form.

I'm kinda stuck on this one.

Before I mention what I've tried, let me say what the book's answer for this one is:

$\displaystyle (2 - \sqrt{3})^{12}$

My plan of attack was to write $\displaystyle \sqrt{3 - i}$ in rectangular form, giving me

$\displaystyle \sqrt{3 - i} = 10^{1/4}~cos \left ( \frac{1}{2} tan^{-1} \left ( -\frac{1}{3} \right ) \right ) + i \cdot 10^{1/4}~sin \left ( \frac{1}{2} tan^{-1} \left ( -\frac{1}{3} \right ) \right )$

(By the way, this approach doesn't use that there are actually two values for $\displaystyle \sqrt{3 - i}$. I'm not sure which one to pick, if it even makes a difference.)

Now that I've got that, then I can rewrite the original problem:

$\displaystyle \left ( 1 - \frac{\sqrt{3 - i}}{2} \right ) ^{24} = $$\displaystyle \left [ 1 - \frac{10^{1/4}}{2}~cos \left ( \frac{1}{2} tan^{-1} \left ( -\frac{1}{3} \right ) \right ) - i \cdot \frac{10^{1/4}}{2}~sin \left ( \frac{1}{2} tan^{-1} \left ( -\frac{1}{3} \right ) \right ) \right ] ^{24}$

Now, believe it or not I can actually do something with this, but it's too complicated for me to bother with the LaTeX. Suffice it to say I can put the base in trigonometric form and use De Moivre's theorem to finish up the problem. But the answer is inordinately messy and I'm frankly not sure I got all of it right with all the picky details to keep track of.

And it certainly isn't the nice form $\displaystyle (2 - \sqrt{3})^{12}$.

Any thoughts as to how to approach this? Thanks!

-Dan