Question

Use De Moivre's theorem to find the power of (2-2i)^5. Answer needs to be in standard form.

Answer 1

Well, we need to turn the original portion, the 2-2i part, into polar form first. Do you know how to do that?

Answer 2

No

Answer 3

Alright, we need to know a couple of conversions then. So the form we want is this:
\[r(\cos \theta + isin \theta)\]This means we need conversions togo froom rectangular numbers to radius and angle. So these are the two conversions we need:
\[r=\sqrt{a ^{2}+b ^{2}} \]which you may recognize as a form of pythagorean theorem. The second we need is:
\[\tan \theta =\frac{ b }{ a }\] In this problem, a is 2 and b is -2. So using a = 2 and b = -2, do you think you could get me r to start out with?

Answer 4

r=sqrt(8) tan=-1?

Answer 5

Right, square root 8, which we can simplify to 2sqrt2. And we have tan(theta) = -1. We still need to solve for theta.

Answer 6

-.785

Answer 7

Well, we want to try to avoid decimals as much as we can. It's usually a good habit to keep numbers as regular degree or radian measures. Do you know what that is in regular radian measures?

Answer 8

I don't remember how to find it.

Answer 9

Well, it requires some unit circle knowledge. For all trig stuff its best to know the unit circle or have a chart handy. So for us the radian measure we need is 7pi/4. This means we have this for our polar form:
\[2\sqrt{2}(\cos \frac{ 7\pi }{ 4}+isin \frac{ 7\pi }{ 4 })\]Do you understand how we got that?

Answer 10

This is my understanding. That whole formula is in polar form. 2sqrt(2) is r. I found a chart and I think I understand how you found the radians now.

Answer 11

That's good :3 Its just remembering the conversions is all. Okay, now that we have this form we can get our 5th power. This is the formula for it.
\[r ^{n}(\cos(n \theta)+isin(n \theta))\] n is the power we want. So what this says is we raise r to the 5th power and we multiply our angles by 5. Think you can do that part?

Answer 12

Stay away from decimals best you can, its definitelty best to avoid decimals when possible.

Answer 13

128sqrt(2)(cos 35pi/4 + isin 35pi/4)

Answer 14

Awesome :D So that would be your answer :P

Answer 15

Unless it wants it in rectangular form, since this is polar form xD

Answer 16

Thanks, until now, I had barely any idea what de moivres theorem was.

Answer 17

He has like 50 theorems. A lot of the mathematicians who come up with the things we use have a billion theorems xD But yeah, np :3.