In Exercises 53–64, use DeMoivre's Theorem to find the indicated ...

Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form.   _(√3 − i)⁶

Accepted Answer

Welcome back. I am so glad you're here. We are asked to find the value of the given complex number using De Mavers theorem. Express your answer in rectangular form. Our given complex number is the quantity of one minus square root three. I all race to the 16th power. Our answer choices are answer choice. A 32,768 plus 32,768 square root three. I answer choice B negative 32,768 minus 32,768 square root three. I answer choice C negative 32,768 plus 32,768 score at three I and answer choice D 32,768 minus 32,768 square root three. I, all right. So we've got this number here and we recognize that we have inside of the parentheses. A number that is in rectangular form that's in A plus B I. And then notably, it's raised two for Demore's theorem, we consider that 16 to be the N. So our N here is 16 step one to figure out the value of this using demo theorem is to convert the part that's in rectangular form, the A plus B into polar form. So let's start by doing that. Let's identify our A and our B, our A here is one and our B is negative square root three. So if we imagine for a moment that we've got a complex plane, we have a vertical imaginary axis and a horizontal real axis. And then we have our coordinates. A is along the real axis and that's moving one unit to the right and B is along the imaginary axis axis. Since that's negative, we're heading down into quadrant four. So notably, we're talking about quadrant four and we'll come back to that. Now getting this into polar form, that's going to be R multiplied by the quantity of the cosine of theta plus I sine theta. And the whole time this will still be raised to that end value. And so in order to convert from rectangular form to polar form, we know that R equals the square root of A squared plus B squared. And we'll start with that one since we know A and BR equals the square root of A is one squared plus B is negative square at three squared. Simplifying this one squared is one. So we've got the square root of one plus negative square root three squared is just +31 plus three is four. So this is a square root of four which is two, our R value is two, we can fill that in. And now figuring out theta, we know that the tangent of theta equals V divided by A. So B again is negative square root three divided by A which is one. So here, the tangent of theta equals negative square root three. Well, we can think for a moment, we recall from previous lessons and from the tables in the textbook that the tangent of theta equals a positive square root three in quadrant one, when we're talking about pi divided by three. So if we know that we're in quadrant four, our angle for the for the tangent of theta is going to be where theta equals five pi divided by three pi divided by three is the reference angle for five pi divided by three. So now we can fill that in. So we've got two multiplied by the quantity of the cosine of our theta is five pi divided by three plus I sine of five pi divided by three. And that whole thing is raised to the end which is the 16th. Now here, now let's talk about Demore's theorem. So this whole thing our are multiplied by the quantity of the cosine of theta plus I sine theta. All raised to the N in De Mavers theorem is equal to R raised to the N multiplied by the quantity of the cosine of theta plus or excuse me missed something. OK. So it's cosine N multiplied by theta plus I sign of N multiplied by theta. So now we can just plug things in, we know our Rs and Ns and theta. So R is two raised to the N and is 16, multiplied by the quantity of the cosine of N is 16, multiplied by theta is five pi divided by three plus I sine of N is 16 multiplied by theta is five pi divided by three. OK. Now, now it's all just simplification to raise to the 16th is 65,536. Multiply by the quantity of the cosine of 16 multiplied by five pi divided by three is going to be 80 pi divided by three. And then plus I sign of the same thing. 16 multiplied by five pi divided by three is 80 pi divided by three. Now, you want to make sure that your calculator is in radiance for the next part bringing down our 65,536 multiplied by the quantity of in your calculator. If you figure out the cosine of 80 pi divided by three, that's a negative one half plus. Now, the sine of 80 pi divided by three, that's going to be square root three divided by two. And don't forget the I now distributing our 65,536 multiply that by negative one half, we get negative 32,768 and 65,536 multiplied by square root three, divided by two is plus 32,768 square at three. And don't forget the I and there it is, there's our given complex number. We look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. _(√3 − i)⁶

Key Concepts

DeMoivre's Theorem

Polar and Rectangular Forms of Complex Numbers

Modulus and Argument of Complex Numbers

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