In Exercises 81–86, solve each equation in the complex number sys...

Question

In Exercises 81–86, solve each equation in the complex number system. Express solutions in polar and rectangular form.

x⁴ + 16i = 0

Accepted Answer

Welcome back. I am so glad you're here. We're asked to represent the solutions in polar and rectangular form. After solving the given equation in the complex number system, our given equation is X rays to the fourth power plus 625 I equals zero. And we're solving for X here. So what we want to do is start off the way we would solve for X trying to get it by itself on one side of the equal sign. So we'll subtract 625 I from both sides of the equation. And then we'll have X rays to the fourth power equals negative 625 I. And then to solve for X, we would take the fourth right of both sides. But we recognize on the right hand side, we are taking the fourth rut of a complex number of negative 625. I. So we want to pause and recall from previous lessons. How do we take a rut of a complex number? So the first thing we wanna do is convert it to polar form. We recall from previous lessons that polar form is R multiplied by the quantity of the cosine of theta plus I sine theta where R equals the square root of A squared plus B squared. And we find theta by knowing that the tangent of theta equals B divided by A given our complex number in standard form of A plus B I. And so we do have it in standard form. It just doesn't really look the way it normally does. In standard form, we have negative 625 I, which is the same as zero minus 625. I, so our A here is zero and our B is negative 625. So now we can fill in A and B to sol for R and theta, let's start with R, we're taking the square root of A squared, that's zero squared plus B squared. B is negative 625 and we'll square that. Well, zero squared is zero. And if we take negative 625 squared and we'll get a positive number and we take the square rut of that, then what we're doing is we're going to get a positive 625 for our R. Now, for theta, we've got the tangent of theta equals B which is negative 625 divided by A, which is zero. But that's undefined. Well, when is the tangent of the undefined? Well, it's actually undefined on some quadrantic angles and we can discern which one because we know that for A, that is our horizontal component and B is our vertical component. We know that A is positive and B is negative. So that's going to be for our quadruple angle of three pi divided by two. So when theta equals three pi divided by two tangent is undefined, so we can plug R and theta in for our polar form we've got for R 625 multiplied by the quantity of our cosine is the cosine of three pi divided by two plus, I multiplied by the sign of three pi divided by two. And now we want to recall that that's the same as negative 625. I. So what we're doing is we are taking the fourth rot of that. So now we've got it in polar form. We recall from previous lessons that will take the root by using Demaris theorem. So for demos theorem, we want to identify our N, which rot are we taking? We're taking the four thre so our N value is four and that tells us we're going to have four solutions here. So we can put for our X sub zero. Well, I'll use Z as they do in the textbook for our Z subzero. We'll have a Z sub one, A Z sub two and A Z sub three. And following the steps in the textbook for our Z subzero, we're going to take the N th threat of our R. So here we're taking the fourth threat of 625 that equals a positive five. And that's multiplied by the quantity of, for our Z subzero. We're going to take our argument. That's our, theta, our three pi divided by two and divide that by our N which is four. And so that's going to be three pi divided by eight. So we'll have five, multiplied by the quantity of the cosine of three pi divided by eight plus I sine three pi divided by eight. And there's our first one, our Z subzero. Now to find the subsequent ones, we're going to add our three, we're going to add a full rotation which since we're in radiance is too high divided by our N which is four and two pi divided by four, simplifies two pi divided by two. So to find a disease of one, we'll take our argument three pi divided by eight and add that rotation divided by N. So adding pi divided by two and to get a common denominator, we'll multiply that pi divided by two by four fourths. And so we're adding three pi plus four pi which gives us seven pi. So we'll have five, multiplied by the quantity of the cosine of seven pi divided by eight plus I sine of seven pi divided by eight. There's our Z sub one. Now we're going to take seven pi divided by eight and add that full rotation divided by N. So we're adding with the common denominator eight or excuse me, not 84 pi divided by eight and seven pi plus four pi is 11 pi. So we have five multiplied by the quantity of the cosine of 11 pi divided by eight plus I sine of 11 pi divided by eight. And we'll take this last 1, 11 pi divided by eight and add four pi divided by eight again. And that'll give us 15 pi divided by eight. So where Z sub three is five multiplied by the quantity of the cosine of 15 pi divided by eight plus I multiplied by the sign of 15 pi divided by eight. And there are our ruts are solutions for X in polar form. Now to get those to be rectangular form, essentially, we're just going to distribute that five to the exact values of the numbers inside of the parentheses. Now, these are not on the unit circle. So I recommend using a calculator. If you take five multiplied by the cosine of three pi divided by eight and make sure that you're in radiance, you will get a positive 1.9 then you take five multiplied by the sign of three pi divided by eight and you get a positive 4.6. And don't forget that's being multiplied by I for the 2nd 15 multiplied by the cosine of seven pi divided by eight. You get a negative 4.6 and five multiplied by the sign of seven pi divided by eight, you get a plus 1.9. Don't forget that's multiplied by I. Now, for our third solution, five multiplied by the cosine of 11 pi divided by eight is a negative 1.9 and five multiplied by the sign of pi divided by eight is a negative 4.6. Don't forget the I and lastly five multiplied by the cosine of 15 pi divided by eight is positive 4.6 and five multiplied by the sign of 15 pi divided by eight is a negative 1.9. Don't forget the I and there are our solutions in rectangular form. Well done. We'll catch you on the next one.

In Exercises 81–86, solve each equation in the complex number system. Express solutions in polar and rectangular form. x⁴ + 16i = 0

Key Concepts

Complex Numbers

Polar and Rectangular Forms

Roots of Complex Numbers

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