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

Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√2 (cos (5π/6) + i sin (5π/6))]⁴

Accepted Answer

Hello, today we're going to be using Demon's theorem to find a simplified form of the given complex number. So there are a few things that Demo's theorem states one, we have a given complex number in polar form which is defined as Z is equal to R multiplied by cosine theta plus I sine theta. And Demo's theorem states that if you want to race this complex number to a given exponent N and N is a positive integer, then Z raced to the power of N can be rewritten as R to the power of N multiplied by the quantity cosine of N multiplied by theta plus I multiplied by sine of N multiplied by theta. So let's take a look at the given complex number. The complex number given to us is the square root of six multiplied by the quantity cosine of two pi over three plus I multiplied by sine of two pi over three. And this entire quantity is raised to the power of four. So in order to use the Mov theorem, there's three values, we need to identify, we need to identify our coefficient of R the angle theta and the exponent of N, which is a positive integer. So let's start by identifying our end value. Our N value is the positive integer that the complex number is raised to. And in this case, it's going to be four. So N is going to equal to four. Next, let's identify R and R is going to be the coefficient in front of the complex number which in this case is going to be the square root of six. So R is going to equal to the square root of six. Finally, we need to identify theta and theta is the angle given in the signing cosine values which in this case is going to be two pi over three. So theta will equal to two pi over three. Now that we have the three given numbers, we can use Demo's theorem to state that Z raced to the power of four will equal to R race to the power of N which will be the square of six. Raced to the power of four multiplied by the quantity cosine of N multiplied by theta, which will be four multiplied by two pi over three. Then we'll be adding I multiplied by sine of N multiplied by theta which will be four multiplied by two pi over three. Now, what we'll need to do is we'll need to simplify the given statement. So the square of six raced to the power of four will simplify to give us the value of 36 next four, multiplied by two pi over three will give us the value of eight pi over three. And this will be true for the angle of sine. Since it uses the same values. Now, we'll need to simplify cosine eight pi over three and sign eight pi over three. But one thing to note is that eight pi over three is greater than two pi. So we'll need to find a co terminal angle. And in order to get the co terminal angle, we'll need to subtract two pi from eight pi over three, we can rewrite two pie as six pie over three. And we have eight pi over three, subtract six pi over three. This will simplify to leave us with two pi over three, which is the co terminal angle for eight pi over three. So we can rewrite the Mavs theorem as 36 multiplied by the quantity cosine of two pi over three plus I multiplied by sine of two pi over three. What we'll need to do is we'll need to simplify cosign and sign of two pi over three. And we can do this by using a unit circle. So we'll first need to identify where two pi over three is located on the unit circle. Two pi over three is located in the second quadrant. And the terminal point at two pi over three is given to us as negative 1/2 comma square root 3/2. And recall that cosine represents any X value of the terminal points and sine represents any Y value of the terminal points. What this means is that we can rewrite cosine two pi over three as negative 1/2 plus I multiplied it by sine two pi over three, which can be rewritten as the square root 3/2. Finally, we can simplify this by distributing the value of 36 into our complex number. And 36 multiplied by negative 1/2 will give us the value of negative 18 and 36 multiplied by the square root 3/ will give us a final value of 18 square root of three multiplied by I. And this will be the simplified form of the complex number using Demo's theorem. And with that being said, the answer to this problem will be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√2 (cos (5π/6) + i sin (5π/6))]⁴

Key Concepts

DeMoivre's Theorem

Polar and Rectangular Forms of Complex Numbers

Trigonometric Identities for Angle Multiplication

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