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Use (14) to find the value of the given principal $n$ th root function at the given value of $z$.$z^{1 / 4}, z=-1+sqrt{3} i$...

Question

Use (14) to find the value of the given principal $n$ th root function at the given value of $z$.$z^{1 / 4}, z=-1+sqrt{3} i$

Use (14) to find the value of the given principal $n$ th root function at the given value of $z$.$z^{1 / 4}, z=-1+sqrt{3} i$



Answers

Evaluating for Given Values using $x=3, y=4,$ and $z=-1$
$$\sqrt[4]{x^{3}+14 y+2 z}$$

Okay, so we want to find the solutions of this equation Z cube money for every three months for I equals zero. And I'm taking the liberty of movement, everything to the right side except for our ze term. So the first thing we do is find our, which is square of a plus B squared. And that's just a book, too, for three. Because that's our A turn for three. Where plus square, that's all under there. Adequate. And that's equal to bury of 64 Of course. And then we're gonna need to find the and you can use this equation can Data will be over a He is four, and A is for times three. And this is just gonna be won over 33 and then you can take the arc. Tanne both. So you are tan won over everything. Cool beta and our tent of one of the Route three. We know where in the first Waldron is. Both A and B are positive that this is gonna be high over six. Okay, so now that we have both are and data we can't use our roots equation. So first que value be for zero we're going to do for 01 and two because our M people to three on the first slide, the cable's era would have the third rule of our it was eight Texaco sign. Hi. Six plus zero birthing. All right, sign Not too, because the three of a two I'll be two times sign pi over 18. What? I Sorry. Hi. No fairy cake was warm equation we're gonna have too, right. It's do this time. So Cuba to be and co sign over six this time plus two pi. I sign pie on six plus two. All over. This is will be equal to two kinds of co sign 13 pie a team. All right. Over 18. I sorry. 13. Hi. Over 18. Under simplifying what was in this complicated expression you can sew for our last crazy in K equals Sue. We're going to who sign? 664 All right. Okay, I see pie on six. Plus high over three. That's two two co sign 25 high. This is simplifying 18 but I sign 25 high over 18. There's aren't the solution

All right. We want to solve the equation. Zeke. You plus one equals negative. I Well, the first step is easy. We want to solve the equation. Z cubed equals negative one minus I. Okay, but then what are the complex numbers who have that as they're cute? Well, let's get a sense of it here. So negative. One month's I is right here on the complex plane. It's got a radius of route to where I can use wth e I thank you and cheer him on one and one to find out that that's rude, too. And I can also pretty easily see that it's got an angle off. 2 25 right? 1 80 plus that extra 45. It's got an angle of 2 25 All right, so I know that the key routes then are going to be have a radius of the cube root of the square root of two. A nicer way of saying that would be the six through two. And the angles are going to be well. The 1st 1 is to 25 divided by three, which is let's see 75 and then the other ones are going to be, Ah, 120 degrees away from that. So the next one is gonna be with an angle of that plus 1 20 So 1 95 and then the last one is gonna be that plus 1 20 again. So to no side. Three. 15. Yes, 3 15 Okay. Oh, make Oh, my gosh. What'd I just say? The radius is six through to the angle is 3 15 That's what I meant to say. OK, so anyway, what are those? Ah, written out in full? That's just a sort of shorthand version written out in full. The roots are I'll call him W one. Don't be too. And w three, these Air Three solutions to that equation. We've got the six through two times co sign of It's 75 plus I sine of 75. And then we have the sixth rid of two times co sign of 1 95 close by sign of 1 95 And then we have the 632 times co sign of to Sorry. What was it? All right, 3 15 plus I sine of 3 15 So those three numbers are the solutions to this equation. Zeke, you plus one equals negative. I These are the three solutions

So to start this problem, we can go ahead and say that, Um since we know that negative to minus two eye is 1/4 root of Z, that means that Z to the 1/4 power is equal to negative to minus two I. So to get rid of this exponents, I'm going to raise both sides to the fourth power and we get Z is equal to negative to minus two I to the fourth power. And so from here, we know that we can apply. Um, the mob serum Thio calculate, uh, what this is going to be. And so we know that we need Thio first, find Orrin data, uh, as well as n so EnergyStar exponents, which is four in this case. So we can go and write that down right away. Are is going to be the square root of a squared plus B squared and a beer both negative too. So are you going to the square root of eight? And data is going to be the inverse tangent of Be over a So when we take the inverse tangent of one that's going to have both a negative sign and co sign value. We're going to find that theta is equal to five pi over four. So when we plug our numbers into dim long serum, um, we find that Z is equal to our to the end. Power so scared of eight to the fourth Power times the coastline of end times data. So and it's four, and that is five pi over four. So and data is just going to be 55 Um, plus I times a sign of the same angle. So I signed +55 and, uh, this word of eight to the fourth power is going to become 64. The coastline of five pie is negative one, and this is going to be plus I time zero. So we find that Z is equal to your negative 64. And so from here we want to find the other fourth roots of Z. So to do so, we are going to use ours e that we just calculated as our A and plug it into this formula over here for finding the route. And so we know that we need to find, um, are and data and K to apply this formula here. So are is equal to we said, the square root of a squared plus B squared, which in our case, is just going to be positive. 64. Um, data is going to be equal to, um, the inverse tangent of Be Over a, which here will give us the inverse tangent of zero over a negative coastline value. So that means that data is equal to pi. Um, since we're finding the fourth Roots, we know that and is equal to four. And finally, our K values are going to start at zero and go all the way up through n minus one, so they will be 012 And and so from here, we can go ahead and find our fourth roots by plugging into you this formula that I drew a red arrow next to. So our first route, um, is going to be, ah, fourth root of 64 which is actually just the square to eight times. We're going to plug in zero for kay here. So it will be the co sign of, um, fatal, which is pi over and which is for plus I times the sign of the same angle before and when we simplify. Um, this will reduce to two plus two I and then to calculate the next route we can essentially do the same thing here. Um, are coefficient will still be the spirit of eight. But this time we're going to have because sign of unplugging it and plugging in one for rk value. So that'll be pi plus two pi, which gives us three price over r n a four. Um and then plus, I'd has the sign of the same same angle. So three pi over four. And when we simplify this, we are going to get this is equal to negative too. Plus two I and we could do the same thing for each of our other two routes on C three and C four, where we will plug in K equals two and K equals three, respectively. And so we find that our other two routes are negative to minus two I and positive to minus two. I

So hopefully with the practice you're starting to get the thing hang of things Here we are going to start by squaring this holy question because we see the square roots on both sides of the equation. So on the left hand side, we have four z plus one is equal to and here we needed again think about that distributive property. So nine plus three square root of four Z minus two plus another three square root of four Z minus two and then plus for Z minus two. We found this by the distributive property of this quantity here times itself eso Then we're just gonna start combining like terms and moving to some terms to the other side so that we are isolating the square root on this side. So on this side, we're gonna have six square root of for Z minus two. And that's from combining these two terms here in the middle. On the left hand side, we're going to be left with the four disease will cancel. We have a negative to in a positive nine. So that leaves us with seven on this right hand side. And when we subtract that over that leaves us with a negative six. Our next step here would be to divide when we divide. By that six were left with negative one equals the square root of four Z minus two. We can actually stop solving right here because we know that a square root when we take the square root of a number that can never equal a negative number. So right from this piece of evidence here, and you could actually have even looked at this step here. Um, we know there is no solution or the solution to this is an empty such she may see that written as, um, this symbol here, which means there's no solutions.


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