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

Question

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

The complex sixth roots of 1 + i

Accepted Answer

Welcome back everyone in this problem, we want to determine all the complex routes for the given complex number one plus the squared of three. I, we want to write our complex roots in rectangular form and express our answer in one decimal. If required note that for our answer choices, there are basically variations of the numbers 1.1 and 0.3 where some are either positive or negative and some are either real or imaginary. Now, what do we really want to know? As I said, we want to find the complex roots but to find the complex roots, we need to use the more various theorem. But to use the theorem, our numbers have to be in polar form. Our number here is already in rectangular form. So we're going to have to come up with a game plan. So first we're going to have to convert, let me put that in black ink is we're going to have to convert our number to pull our farm. OK. After doing that, that is when we can use the Morris theorem to find a route to find the en route. But remember that our answer has to be in rectangular form. So we have to rewrite after we've done that. After we've found that route, we'll have to convert our answer back to rectangular form. So we have a few stuff to do, right? So let's start with the first one, let's convert our, our rectangular form of our complex number to polar form. Now, I recall that for a rec complex number in rectangular form, OK. Then in polar form, that number would have been R multiplied by the cosine of theta plus I multiplied by the sign of theta where R equals the square root of X squared plus Y squared. And theta equals to the arc tangent of Y divided by X. So we can substitute our values because notice that the value of X is one and the value of Y is the square of three. So we can substitute those for R and theta namely R is going to be one squared or the square root of one squared plus the square of three squared, which when we were that old one squared is one, the square of three squared is 31 plus three is four and the square of four equals two. So R equals two. Likewise our angle theta will be the a gent of Y which was the spirit of three divided by X which was one and the skirt of three divided by one is a skirt of three, the arc tangent of the skirt of three equals degrees. Therefore, in polar form, our number would be written as two multiplied by the cosine of degrees plus I multiplied by the sign of degrees. So we have converted our number to polar form. No, we can use the Marvis theorem adjusted for the root. OK. Because recall that demo said if we have a complex number written in polar form, then the complex route are each complex route would be equal to the root of the value of R multiplied by the cosine of the angle. Theta plus 360 degrees multiplied by K divided by N plus I multiplied by a sign, the sign of the same angle, theta plus 360 degrees K multiplied by K divided by N right, where K would be all of the values leading up to N minus one. So 012, all the way to N minus one. So we already have these values right? Because for one, we know that we want a complex fourth root. So that means N equals four. OK. We also know that our value for R in our complex numbers too and our anger is 60 degrees theta is 60 degrees. So we can substitute into our theorem for us to come up with a formula to find all the roots of our complex number. Because this will not imply that each case where for KB equal to 012 and three would be equal to the fourth root of two, multiplied by the cosine of degrees plus 360 degrees multiplied by K divided by N plus I multiplied by the sign of that same angle. 60 degrees, 360 degrees multiplied by K divided by N four. The values of K being equal to +012 and three. Remember K goes up to one less than N. So since N is four, the last value of K would be three. And I'm going to simplify this before we work because we can change our values of N as we said, N equals four. OK. And if we put that into our formula, now the fourth root of two, I'm going to leave it that way and I can simplify the angle because 60 divided by four is 15 degrees and 360 degrees multiplied by K divided by four would have been 90 degrees. And I'm going to put K in a little, let's say it's own little bubble for no, just to make things easier while we work. Because you'll notice that as we go along, the only thing that's going to be changing is the value of K and the same angle would go for I multiplied by sine. OK. So no, what we're going to do is we're going to change out those values of K as we go along. So let's do that. Let's do that. For each case, I'm going to do the first and then I'm going to allow you to work on the second, the, the, the second, the third and the fourth and then we'll compare what we have. So for our case four K equals zero, then our complex number is going to be the fourth root of two multiplied by all of what we have here. Do you remember what I was telling you? I'm going to make it a little easy. So I'm just going to copy and paste this. I, I really hope you can copy and paste it too, but I'm going to do that because as I told you, the only thing that's going to be changing from here on in is the value of K. So no, all I'm going to do is replace my values of K with the value that I'm working with in this case, zero. Now the first route of two was going to be multiplied by the cosine of 15 plus 90 multiplied by zero, which would just be the core sign of 15 degrees and I multiplied by the sign of the same angle 15 degrees. Now we're almost there. But remember we need it in rectangular form. So to write this in rectangular form, all I need to do is to multiply or expand what I have in my brackets. So that will be the fourth root of two multiplied by the core sign of degrees, which would be 1.1 and the fourth root of two multiplied by, I multiplied by the sign of 15 degrees, which would give me plus 0.3. I, so that would be the first of our four roots. No, we just need to figure out the rest. Let's go ahead and try and figure it out for when K equals one. Then our complex number would be equal to negative 0.3 plus 1.1. I notice that I just changed the values of K and went ahead and worked them out 15 plus 90 degrees equals 100 and five degrees. And I would multiply through by R as usual. Now, we're going to find to do the same to find the values or the roots when K equals one, sorry K equals two and K equals three. When K equals two, sorry. When K equals two, I should say then my rectangular uh my form, my answer in rectangular form eventually works to negative 1. minus 0.3. I finally, when K equals three, then I replace my values of K for three. And our rectangular form eventually ends up being 0. minus 1.1. I therefore, these are all our possible answers. The, these are the complex fourth roots of one plus the square root of three. I. And if we go back to our answer choices, you'll realize that that one was answer choice. D guys guess what? This was a big one. You did a really great job. Thanks a lot for listening. I hope this video helped.

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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 sixth roots of 1 + i