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Find and graph all specified roots of 1.$$ ext { fourth }$$...

Question

Find and graph all specified roots of 1.$$ ext { fourth }$$

Find and graph all specified roots of 1. $$ \text { fourth } $$



Answers

Roots of Complex Numbers Find the indicated roots, and graph the roots in the complex plane.
The fourth roots of $-1$

Okay for this problem. They want us to plot the complex fourth roots of one. So first we have to remember that as a complex number, one would be graft going out the rial axis to a value of one. And so that has a module, lists or radius of one and an angle with the positive real axis of zero. So we're talking about one being in polar form one times co sign of zero degrees, plus I times a sign of zero degrees or radiance. Either way, uh, zero degrees, zero radiance. That one doesn't matter. So we would have to use demo of race, the room here where we need the different values for Kay from our formula. And in this case, since we're doing the four through, there are four values of K starting at zero and going up to four minus one, which is three. So K equals zero. When we use our formula, we have the end through love one which is one times the co sign of our angle zero divided by, and the index on our roof zero divided by forced zero. Plus I turns a sign of zero. And then these other angles air made by adding on the 360 divided by N or the two pi divided by and each value for each value of K. So 3 60 divided by four is gonna give us 90 degrees. So we just add 90 degrees each time to get the next route that we have so co sign of 90 degrees. Plus I sine of 90 degrees is our second route. Then we had 90 again. And so we end up with the co sign of 180 degrees, plus a high times a sign of 180 degrees. And then one last time adding 92 the 1 80 we get to co sign off 270 degrees plus high times a sign of 270 degrees. And now these have exact values, and we can put these into rectangular form. Using that co sign of zero is one sign of zero is zero. So that equals one co sign of 90 0 sign of 90 is one. So that gives us I co sign of 100 eighty's negative one sign of 100 and 80 a zero, so that gives us negative one as that route and then co sign of 270 a zero. Hi. Time to sign. Up to 70 is negative one. So that gives us negative I So now when it comes to plotting these one, we already know where that ISS it's there. On the real axis, I would be up to one on the imaginary axis. Negative one would be over to negative one on the real axis, and then negative. I would be down to negative one on the imaginary axis, so those

They have 78 number problems which we need to find the cube groups off. Four Rule three Bless for character. So in polar form this converting as their equal to 16 48 course by where six place I after sane Bye bye six. Okay, Okay. No, we need to find it. Que route. So we have to find one by three one by three This will be called toe 80. Depart one by three Course to get by. Bless Bye bye six by three Bless. I, uh, sign to give by. Let's buy a Wessex way three. Let us assume this to be that k. This conversion is a tree 12 1 by three is gay course toe give by by three bless Bye bye 18 blurs I hossein Okay, bye bye. Three less by by eating now of NK koru zero first you brood will be Quito Do of course by away 18 Bless Iron Hossein by writing. This is the first you build When k call to one is that one will be equal toe dio Of course this will be to buy where three so 253 Because Dubai where three plus by by 18. So 12 plus one turned in by late in less tired assign. I'm hurting by way 18 third. Cute. That would be equal to do course. Fought by by 3 24 25 Pilot 18. 25 by by 18. Plus, I would assign 25. Bye bye. It didn't. So these are the cube groups. Now we will be drying these groups, so let's get the next plane. Oh, yes. This is a complex thing, Finch, the cereal bar decision and that he exists. And this is the the Lexus and all of the complex numbers has more political due to. So we will be trying first the circle with the Radius, too, that in his, um, days to the circle of radius too. So this the first will be bye bye, 18 angle. So by write in somewhere here this is his that Tzeitel, then turning by my 18. So starting by reading somebody is okay? No, it will not be somewhere here. 13 by where 18 will be somewhere here. This is said one. And that too is 25 by way it in. So Byblos seven by 18 It's somebody here. Is it to think of some

You have 18 number problem in which we need to find Ah, fifth root off 32 50 Route off 32 1st of all, let us assume that to be equal to 32 that is 32 players. The writer so in polar form is that it will be equal to simply started toe because it'll plus I wrote that Signs it up So we need to find you stayed through won by five So one by fight Let us his young days Toby, is that okay? 32 aerated part one by five course zero plus two K pirates report If I do have a fire blurs iron does sign a zero place to keep I divide by five This will be simply too course to deep by by five blurs iron They're sane to get by by five Now let us like in the values off k one by one when cake with zero there is zero will be equal to two cause zero less I wrote a sign zero que call to one That one will be equal toe toe course to buy where five plus I would assign do by by five of an cable. Due to that that is heard. Fifth root two course for by my fire. Plus, I had a sign for by well, five when cable to full on 33 Que called three is that people we call toe Toe cause 655 No, sire. Tusayan 6.5 when k called before, That's one will be able to do course, eat by my fire nurse. I just saying by five. So these are the roots now. We will be during these grooves on the complex plane. So that is good. Complex plane. Oh, okay. Very good. Okay, Now you can see that all have more or less equal to two. So we're drawing a circle with Really is to first. This is you get access the city. Many exes do do two and two. So just how clever it is, too. The 1st 5th through days, zero zero. So yeah, it will be. This is their zero. Is that one is Dubai where? Five. So somebody is the zed. One is that too, is for by their five that is 255 for private five. Somewhere here that to is that freeze six Piper five. So somewhere here is that three. And that four is eight by over five. That is some very here, that for thank

We have problem. But even in which we need to find a throwed off one, Let us assumes article 21 So one place, zero ETA So in polar form, this will Britain is off one cause zero place I read that sign zero Now we need toe find a throats one by eight So let us say the on days Tau bay that K will be equal to cost Developers don't get higher by eight That is to give by by eight blurs I order sign to get by eight It is cause gate by by four plus I wrote a sane gate by where for now we will be playing in the values of K one by one cake or zero So first, a true it will be cause zero plus Arte sign Zito then cake or to win as Evan will be second to a second eight through So it will be, of course, by by four blows I order sign I live for when k called to that will be called toe cause to buy. Before that, it's bye bye to plus our hossein By what? Yeah, a friend cake Or to three. Is that three will be called Toe course 354 Bless. Hired a sign treat my way for when cake Alito four That for will be cause by Plus I are the same by of an cake although five That five will be course fight by wherefore bless I had a sign fight by love For when k called 06 I have six will be equal toe Of course 654 That is three pi over two. Plus I wrote a sign three by way to when Keiko to seven. That it last eighth route. That seven will be of course, 75 or four. Bless. Hired a sign 75 Wherefore so. These are all the age groups. All the eight route. OK, No, uh, we need to draw these groups on complex plane. So let us get a plane here. We have got it. Okay, e that devotee. But what has every dog has on the more or less equal to one. So we will have to draw a circle off radius one. So let us assume this to be one 1234112344 blocks. Because we have to fit in four or eight groups. Sorry. 88 throws. 1234 We have is the own days to be one. So it is dry. Circle off radius one unit. Okay, so first cube root our first eight through days. Zito Zito here. This is Sarah's itto that won the pie before Said to that three that for is that five is that six and is that seven?


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