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

Accepted Answer

Welcome back everyone. In this problem, we want to determine all the complex roots for 16. And we want to express the complex roots in the rectangular form and in one decimal if required for our answer choices A is 22, I negative two and negative two IB is 16 16, I negative 16 and negative 16 I C is 44, I negative four and negative four I and D is one I negative one and negative I no. What do we want to know? We want to determine the complex roots for our given complex number. And further, we want to find the complex fourth roots of that number. We have the number here 16. And you realize that this is just a real number. There's no complex part. So in complex form, technically, it's really just 16 plus zero. I. Now we know that to find complex roots, we can use the verse theorem. However, the Vez theorem works for complex numbers in polar form. The problem is our numbers in rectangular form. So if we're going to find those complex roots, we need a game plan. First of all, we'll need to convert or number to pull our form. Once it's in polar form, then we can apply the Morse theorem to find the roots. But if you notice in the question, it wants us to express our answer in rectangular form. So after we've applied the theorem, then we'll need to convert our answer back to regular form. A rectangular form. Where did I get regular from? So that's what we'll need to do. That's our game plan. So let's see if we can figure it out. Let's get started by converting our number to polar form. So recall that to convert a number to pull our form, if we have a complex number that's written in the form X plus Y I, then in polar form, the number would be Z equals R multiplied by the cosine of theta plus I multiplied by the sign of theta where R, where R equals the square root of X squared plus Y squared. And theta is equal to the arc tangent of Y divided by X. Now, already, we know that for our number X, our, our real number is 16 and our imaginary number is zero. So we can substitute it in these values to find in these formulas to find the value of R and theta because that means R is going to be the square root of 16 squared plus zero squared, which when we work that out, it just gives us 16 and theta is going to be equal to the arc tangent of zero divided by 16, which would just be zero and the A ton of zero is zero. So since theta is zero degrees and R is equal to 16, then that tells us that in polar form. So in the polar form, our number would be written as multiplied by the cosine of zero degrees plus I multiplied by the sine of zero degrees. So for now we have our number in polar form. No, we can apply the Morse theorem. What does the theorem say? Say? Well, recall that it says for a polar, a complex number expressed in polar form that is I multiplied by the cosine of theater plus I multiplied by the sign of theater. Then for the complex root of that complex number, each case, let me write it on the, on the right, each case will be equal to the nth root of R multiplied by the cosine of or anger plus 360 degrees multiplied by that case divided by N multiplied by K plus I multiplied by the sign of the same angle. So I should write that a little bigger theta plus 360 degrees multiplied by K all divided by N where K equals all of the values leading up to N minus one. So if we can figure out what those values are, if we can substitute, then we can find each route for our complex number. Now, what do we know? Well, here, our complex number is 16 multiplied by the cosine of zero plus I multiplied by the sign of zero. So that tells us that our, so for our number R equals or a theta equals zero degrees. And remember we said we wanted the fourth root of our number. So that means N equals four. So if I plug those values into the more of a theorem for the complex root, then for each case, it will be equal to the fourth root of 16 multiplied by the cosine of zero degrees plus 360 degrees multiplied by K for whichever case divided by N which equals four plus I multiplied by the sine of the same angle, zero degrees plus 360 degrees multiplied by K all divided by four. I can simplify that because the fourth route of equals two, zero, divided by four is 0, 360 divided by four is 90. So this will be the core sign of 90 multiplied by K. I'm going to put K in a special bracket and you'll see why later plus I multiplied by the sign of the same thing because we, it's also zero plus 360 degrees multiplied by K all divided by four. And I'd use this formula to find my complex roots for now, remember K counts up to N minus one. So we'd use it for K equals 012 and three because N equals four So it would be, it would stop at one less than N, which is three. So in order to find each of these routes, all we need to do now is to substitute our values of K into our formula. And then we can solve for each after we've substituted, then we'll be simplifying for the rectangular form. So let's go ahead and do that. I'm going to work through the first one and I'm going to copy my formula to make it a little easier for us to, to continue working it through. OK. So let me just take this right here. But as we go along, remember that all we're doing is substituting each of the values of K. So let's start with K equals zero. So this means that when K equals zero, I replace my K with a zero. And no, I can, I probably didn't need that equal sign. Did I? You know what? Let me keep it. Oops. No, that I've put it right here. I can remove my values of K and no, I can substitute it with zero. So that would be the core sine of 90 multiplied by zero. plus I multiplied by the sine of 90 multiplied by zero. And when I were at the two, the cosine of 90 multiplied by zero is zero. So that's two multiplied by the cosine of zero plus I multiplied by the sign of zero in order to turn this into rectangular form, I can go ahead and multiply note two multiplied by the cosine of zero equals one because the core of zero is one, two multiplied by I multiplied by the sign of zero would be equal to zero because the sign of zero equals one. Therefore, when K equals zero, my complex root equals two. Now let's apply this same idea to figure out the rest of the complex routes. When K equals one, what would my complex root be? No, I'd substitute my values of K for 1 90 multiplied by one is 90 the cosine of 90 is 02, multiplied by zero would be equal to zero. And the sign of 90 equals 12 multiplied by one. I gives us two, I therefore two, I is another root of our expression. Let's move on to when K equals two. No, I multiply 90 by two, which makes me end up with two multiplied by the core sine of 180 degrees plus I multiplied by the sine of degrees. Now the core sign of 180 degrees is negative one and two multiplied by negative one is negative two. The sign of 180 degrees is zero and two multiplied by zero gives us zero. Therefore, our third route is negative two. Finally, let's find the root when Z when K sorry equals three 90 multiplied by three is 270. And the cosine of 270 would be 02, multiplied by zero is zero. The sign of 270 is negative one and two multiplied by negative I would be negative to I. Therefore, the fourth roots of our complex number would be 22, I, negative two and negative two I. And when we look back in our answer choices that would have been answer choice. A great job everyone you did really well. Thanks for watching. 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 sixth roots of 1

Key Concepts

Complex Numbers

Roots of Unity

Rectangular Form

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