In Exercises 69–76, find all the complex roots. Write roots in re...

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In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth.The complex fifth roots of 32

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Welcome back. Everyone in this problem, we want to determine all the complex fifth routes of 1024 expressing our roots in rectangular form and rounding to one decimal if needed. Now, what do we already know? Well, we have a value that we want to find its roots. It's complex roots and we can do that using the theorem. But the problem is our number is in rectangular form and the theorem only works with in number complex numbers in polar form. So you know what we'll need to do first, we need to convert our number to pull our form. Once we've done that, then we can apply the maus theorem. OK. And then once we've used the Ethereum to find the roots, notice our problem wants us to express our roots in rectangular form. So we'll have to convert our not our answers back to rectangular form. So we got a few stuff to do, right? So let's start with the first one. Let's convert our number to pull our form. No. What do we know about that? As a matter of fact, I'm gonna put this in a rectangle just so that we can revisit it. But what do we know about that? Well, we know that for a number in rectangular form recall that is Z equals X plus Y I. Then in polar farm, we can write our number as R are multiplied by the cosine of theta plus I multiplied by the sign of theta. OK. Where R equals the square root of X squared plus Y squared and theta equals the arc tangent of Y divided by X. Now, for our value 1024 notice that there's only a real part so its imaginary part Y would be equal to zero. In that case, that tells us that R equals 1024. And our value theta would be the arc tangent of zero divided by 1 24 which is zero and the arc tangent of zero equals zero. So for our number 1024 in polar farm, it would be equal to multiplied by the cosine of zero plus I multiplied by the sign of zero. So we've done the first part. No, we can apply the more obverse theorem because recall that for a complex number written in polar form, OK. So R multiplied by the cosine of theta plus I multiplied by the sign of theta. Then the, the Marous theorem for the root says that each root of our value, each root would be equal to the N root of R all multiplied by the cosine of theta plus two pi K divided by N plus I multiplied by the sign of the same angle. And this would be equal to all the values of K where K equals 012, all the way up to N minus one. In other words, each route up to one less than our route that we're trying to find. So what does that tell us where we can manipulate this formula? Or we can manipulate the theorem to come up with a formula so that we can find a complex fifth root. No, for it, since we're looking for the fifth roots, so that tells us N equals five R would be equal to 1024 from our, the polar form of our complex number and theta equals zero. Then for each case of our complex route, it would be equal to the fifth route of all multiplied by the cosine of theta, which is zero plus two pi K divided by five plus I multiply it by the sign of the same numbers zero plus two pi K are divided by 54. The values of K being and four. So those are five routes. We can even simplify this some more because the fifth route of 1024 equals four. And in our bracket, the core sign of zero plus two pi K all divided by five that would have just been two pi K divided by five. And the same thing would go for, I multiplied by the sign of two pi K divided by five. So now we have a formula for each case of our complex root. So all we need to do is to substitute the values of K to help us. So for each, I'm going to work the first one through with you and then we're going to work it down. I'll allow you to work the other ones on your own and I'll guide you through it. So let's start for K being equal to zero. So for K equals zero, then we would have four, we're going to replace the K in our formula. All we need to do is to replace it with zero, right? So that would be four multiplied by the cosine of two pi multiplied by zero, all divided by five, which I don't have to write that because two pi multiplied by zero is zero and zero divided by five is zero. So I can just write zero there. So the core sign of zero plus I multiplied by the sign of zero. OK. I should show that that's I, and when we were at that o the core sign of zero is one. So that's four multiplied by one and the sign of zero is zero. So that would be zero, I and no, to write it in rectangular form, all we do is expand across our bracket. So four multiplied by one is 44, multiplied by zero is zero. Therefore, four K equals zero or root equals four. Now, all we're going to do is to try and figure that out for the rest of values for K. So let's go when K equals one. Now when K equals one, then our angle would be two pi multiplied by one which is two pi divided by five plus I multiplied by the sign of two pi divided by five. The cosine of two pi divided by five is 0.309. And the sign of two pi divided by five is 0.951. And when we multiply across, then our answer is 1.2 plus 3.8. I next for K being equal to two, then our root would be or multiplied by the cosine of 4/5 soft pi plus I multiplied by the sign of 4/5 of pi and the cosine of 4/5 of pi is negative 0.809. The sign of 4/5 stuff pi is 0.588. And when we multiply across, we end up with negative 3.2 plus 2.4. I now I'm going to just put in the rest of them and you can compare them to see if you got the same results. So did you get these two, when K equals three then or root becomes four, multiplied by the cosine of six fifth of pi plus I multiplied by the sin of six fifth of pi, which eventually works down to negative 3.2 minus 2.4 I and for K equals four, then our equation becomes four multiplied by the cosine of eight fifths of pi plus I multiplied by the sign of eight fifths of pi which when we work it down becomes 1.2 minus 3.8 I. So our roots for our equation, our complex fifth roots would be 41.2 plus 3.8 I negative 3.2 plus 2.4 I negative 3.2 minus 2.4 I and 1. minus 3.8. I Now, if you've got to this point, you did a great job, really great job. Keep going. Thanks for watching and I hope this video helped.

In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth.The complex fifth roots of 32

Key Concepts

Complex Numbers

Roots of Complex Numbers

Rectangular Form

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