In Exercises 25–29, use DeMoivre's Theorem to find the indicat...

Question

In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. _ (1−i√3)²

Accepted Answer

Hello. Today we're going to be using Demo's theorem to find the value of the given complex number. We're going to be expressing our answer in rectangular form. So the complex number given to us is the square root of 12 minus two. I raised to the power of four. Now, demo's theorem states that if you have a complex number in polar form, which can be written as Z is equal to R multiplied by the quantity cosine data plus I multiplied by sine data. If you have a complex number in this polar form, then if you want to find the value of Z race to the power of N, the value of that complex number will be R raced to the power of N multiplied by cosine of N multiplied by theta plus I multiplied by sine of N multiplied by theta. So in order to answer this problem, we're first going to, to identify our complex number in polar form. Currently, the complex number given to us is in rectangular form and it can be expressed as X minus X plus Y I raised to the power of N. Now, in this case, here we're going to need to solve for R and data in polar form. Let's go ahead and start with R recall that R is equal to the square root of X squared plus Y squared. In this case here, X will equal to the square root of 12 and Y will equal to negative two. So if we plug this in for the value of R, we get the square root of the square root of 12 squared plus negative two squared, the square root of 12 squared will simplify to leave us with and negative two squared will leave us with the value of four, 12 plus four will give us the value of 16. So we have the square root of 16 and the square root of 16 will simplify to four. What this means is that the value of R for a complex polar number is going to be four. Next, let's go ahead and identify our value of data. Now recall that data is equal to the tangent inverse of Y divided by X. This will give us a theta value of the tangent inverse of negative two divided by the square root of 12. Now the square root of 12 can be simplified to four multiplied by the square root of three, which itself can be simplified to two multiplied by the square root of three. So theta is equal to the tangent inverse of negative two divided by two multiplied by the square root of three, we can then simplify this quantity further by simplifying the two in both the numerator and denominator that will give us the as equal to tangent inverse of negative one divided by the square root of three. And if we rationalize both the numerator and denominator, we are left with data is equal to tangent inverse of the square root of three divided by three. And this quantity is negative. Now, what is the value of tangent inverse of negative square root three divided by three? Well, this will give us a theta value of negative pi divided by six. So theta is equal to negative pi divided by six. Now that we have the R and the theta value, we'll need to identify our end value N is going to be the positive integer. That is the exponent given to us. In this case, the positive integer exponent is going to be four. So we're going to allow N to equal a positive four. So just to reiterate, we have our value for R, we have our value for N and we have our value for the, we can then plug in these values into Demo's theorem to find the value of the given complex number. So using these values and using demons theorem, we can state that our complex numbers, Z race to the power of four is equal to four, race to the power of four multiplied by the quantity cosine of N multiplied by theta which be four multiplied by pi divided by six plus I multiplied by sine of four multiply by negative pi divided by six. We'll just need to simplify a few elements here. First four race of the power of four will give us the value of 256 and four multiplied by pi divide. Negative pi divided by six will give us the value of negative two pi divided by three. So we have cosine of negative two pi divided by three plus I multiply it by sine of negative two pi divided by three. Now, in order to simplify this further, we're going to have to rewrite the current value of data in the form of an angle between zero and two pi. In order to do this, we can take negative two pi divided by three and add a value of two pi to this angle. We can rewrite the statement as negative two pi divided by three plus six pi divided by three. And once we add these values together, we'll be left with four pi divided by three. So what this means is that the angle four pi divided by three is equivalent to the angle of negative two pi divided by three. So we can rewrite the complex number as multiplied by the quantity cosine of four pi divided by three plus I multiplied by sine of four pi divided by three. Next, we'll need a unit circle to evaluate the values of sine and cosine. Let's go ahead and draw a unit circle. Now where is the angle four pi divided by three located at? Well, that angle will be located in the third quadrant of the unit circle. And the following terminal point is given to us as negative one divided by two comma negative square root three divided by two. Now recall that any terminal point on the unit circle is defined as cosine comma sign. What this means is that cosine is every X value of the terminal points and sin is every Y value of the terminal points. So if we want to evaluate cosine of four pi divided by three, we'll just have to take a look at the X value of the corresponding terminal point. In this case here, the X value is going to be negative one divided by two. So that means we can rewrite cosine four pi divided by three as negative one divided by two. So we have 256 multiplied by negative one divided by two. Next, we'll need to evaluate sine of four pi divided by three. Now again, in order to evaluate this quantity, we'll just have to take a look at the Y value of the corresponding terminal point. That value will be negative square root three divided by two. So that means we can rewrite this quantity as negative square root three divided by two multiplied by I Now, what we'll need to do is we'll need to distribute the value of 256. Distribute it with negative one divided by two and negative squared three divided by two. Doing so will leave us with negative minus 128 multiplied by the square of three multiplied by I. And this will be the value of the given complex number. 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 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. _ (1−i√3)²

Key Concepts

Complex Numbers in Rectangular and Polar Form

DeMoivre's Theorem

Conversion from Polar to Rectangular Form

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