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AJWGe DY seiccling [re Dest answver cnoice:Question 14Use DeMoivre's Theorem to find the indicated power of the complex number: Write the answer in rectangular...

Question

AJWGe DY seiccling [re Dest answver cnoice:Question 14Use DeMoivre's Theorem to find the indicated power of the complex number: Write the answer in rectangular form;[4(cas 15" +isn 1570] 4128 1281256i128 6 1281428 128-/3

aJWGe DY seiccling [re Dest answver cnoice: Question 14 Use DeMoivre's Theorem to find the indicated power of the complex number: Write the answer in rectangular form; [4(cas 15" +isn 1570] 4 128 1281 256i 128 6 1281 428 128-/3



Answers

Use DeMoivre’s Theorem to find the indicated power of the complex number. Write answers in rectangular form.
$$\left[4\left(\cos 15^{\circ}+i \sin 15^{\circ}\right)\right]^{3}$$

We were asked to find negative the square root of two minus side to the fourth power. And we need to turn this into a pole perform in order to do that. So the first thing we have to do is find the radius, which is gonna be the square root of a squared plus B squared. So that's the square root out square root of two squared plus negative one squared. So that's two plus one. So the radius is the square root of three, and then we have data. The angle that this is making on the graph is gonna be the inverse tangent of be over a negative one over the square root of two. And this one we have to plug into the calculator and we're in Quadrant four. Fortunately, the calculator will give us the Quadrant four answer, and the Catholic was going to show you a negative value for this. It's gonna show you negative 35 0.26 degrees. I'm gonna round it off there. And normally we would use a positive angle on generally that as the negative here, just to save a little bit of work and save some button pushing later on down the road, so that means that our number can get rewritten as the square root of three times co sign of negative 35.26 plus I. Time to sign of negative 35 0.26 and we are raising hope of that to the fourth power. So the module ist the radius gets raised to the power of square root of three to the fourth power, and then the angle. The negative 35 0.26 that gets multiplied by your exponents will multiply that by four. So that will give us in polar form. Square root of three to the fourth power is nine times the co sign of negative 35 1 26 That should be not a five sorry negative 35.26 times four is negative 141 0.4 plus I times a sign of negative 141.4 So there's the pole reform of this, and now we need to find the rectangular form. So that means we have to distribute the nine through. While we were figuring out the sign and co sign of this answer. So nine times the co sign of negative 141.4 It round is going around off to negative seven and nine times the sign of our negative 141.4 round off too negative, 5.66 times.

Okay, We have the complex number 1/2 times the co sign a pie over 12. Plus I sine a pie over 12. And we're raising that to the six power and we're using demon raised here. Um, in order to do that, well, we start by taking the module lists are radius, and we're gonna raise that to the sixth power and then the angle argument, we just multiply by the power. So we're gonna have the pie over 12 time six. So 1/2 to the six power would be 1 64th times three co sign of six times pi over 12 will give us a pie over too. Plus I times a sign. Uh, I over too. And so now we want to turn this into rectangular form, and the co sign of pi over to is zero. And the sign of pi over two is one. And so when we take the 1 64th and distribute it through, we end up with 1 64th times. I

Okay, we have the square to three minus I and we want to raise that to the six power. And I am not a fan of having six. Binomial is all needing to be multiplied together and have to work that out by hand. So we're gonna turn this into polar form and then used anymore Frasier. So the polar form, we need to find the radius, which is the square root of a squared square, three square possum B squared, which is negative. One squared. So that would give us three plus one, which is four. And the square before is too. So there's my radius. The angle theta is the inverse tangent of being negative one over a square or three. Fortunately, this is in quadrant for, so the calculator will give us a direct answer for this, but it's going to give it to us as a negative number. It's going to say negative 30 and normally we use positive angles for this. So we have to take 360 plus. Then I get to 30. So Seita is 330 degrees. I'm using degrees instead of radiance. Doesn't matter which one you want to use. So now that means that we can write this in polar form as two times the call sign off 330 degrees. Plus I time to sign of 330 degrees. And we're gonna raise that old to the sixth power. So the module iss the radius gets raised to the power. So to to the six power and the angle arguments that we have gets multiplied by the power. So we have 330 times six, so two to the six power is 64 times co sign of 330 times six is 1980 which has a few extra spins in it. There's actually five extra rotations here around the axes. So if you subtract off those five extra spins, we're left with ah, 180 degrees as our angle. And so then we have plus I times a sign of 180 degrees. And so now they want this in rectangular form. So we will look up the co sign a 180 which is Nick, if one and the sign of 100 and 80 which is zero. And now it can distribute the 64 through there and 64 times negative one gives us an answer negative.

We have one minus I and we want to raise that to the fifth Power. And I personally would not be a fan of trying to do five of these all multiplied together using the distributive property. So I'm gonna turn this into polar form and then use him or phrase theory to raise it to the power. So we have to find the radius, which is the square root of a squared plus B squared. Hey, is one and B is negative one and so one squared plus negative one squared means our radius is the square root of two. And then data is the inverse tangent of Be over a so negative 1/1, which is in quadrant for so that's going to end up being 315 degrees. Your calculators probably gonna give you an answer of negative 45 degrees. I'm turning it into the positive value just because normally that's what we use when we're doing poor form for complex numbers is positive angles instead of negatives. So that means that one minus side of the fifth Power is the same thing as the square root of two times the co sign of 315 degrees, plus I times a sign of 315 degrees. And that's raised to the fifth power. And now I can apply. Dim over, it's there. So when we do that, we take the module iss the radius and we raised that to our power off five. And we take our angle argument that we have here 315 degrees and we multiply that by five. So is the square root of two to the fifth Power ISS, four times the square root of two, and we have the co sign off 315 times. Five is 1575 which is just a bit more than 360 Fact is, actually has four extra spins in it. So when I subtract off the 1440 degrees for those four extra spends, we end up with That's the same angle has 135 degrees, and then so we have our plus I times a sign 135 degrees. Now they want this in rectangular form, so I need to find the value of co sign and sign of 135 degrees. Well, co sign of 135 is negative one over the square root of two, and the sign of 100 and 35 is positive one over the square root of two. So now I can distribute my four square with the two through my parentheses. And when I do that four square to two times the negative one over square to is negative four and then four square root of two times the one over square to ISS positive for times I


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