In Exercises 5–7, perform the indicated operation. Leave answers ...

Question

In Exercises 5–7, perform the indicated operation. Leave answers in polar form.

[2(cos 10° + i sin 10°)]⁵

Accepted Answer

Hello. Today we're going to be using Demo's theorem to simplify the given complex number. Then we're going to express our answer in polar form. So we are given three multiplied by cosine of eight degrees plus I multiplied by sine of eight degrees. And that entire quantity is raised to the power of seven. In order to find the simplified form of this value, we'll need to use Demo's theorem. Now, Demo's theorem states that you must have a complex number in polar form which can be rewritten as Z is equal to R multiplied by the quantity cosine of data plus I multiplied by sine of data. Now, if you wanted to raise this value to a certain expression or a certain exponent N, as long as N is a positive integer, then Z to the power of N can be written as R to the power of N multiplied by the quantity of cosine of N multiplied by theta plus I multiplied by sine of N multiplied by theta. So in order to simplify this complex number, using Demo's theorem, we're going to need to identify our R value, our end value and our theta value, let's go ahead and start with R. Now R is the coefficient in front of the complex number in polar form. And in the given complex number, that coefficient is three. So we're going to allow R to equal to three next N is the exponent or the positive integer exponent that the complex number is raised to. And in the given complex number that exponent is seven. So we're going to allow N to equal to seven. Finally, theta is the angle inside of the cosine and sine functions. And in the given complex number, that angle is given to us as eight degrees. So we're going to allow theta to equal to eight degrees. Now that we have a RN and the value we can use the MOV theorem to find the simplified form of the given complex number. So Z raced to the power of N which is seven will equal to R raced to the power of N which is three race to the power of seven multiplied by the quantity of cosine of N multiplied by theta which is seven multiplied by eight plus I multiplied by sine of N multiplied by theta which will be seven multiplied by eight. There's a bit of simplification that we can do. Now, three multiplied it by seven will give us the value of 2187 and seven multiplied by eight will give us the value of 56. So we get cosine of 56 plus I multiplied by sign of 56. This is the simplified form of the given complex number in polar form. So this will be the solution to the given problem. And with that being said, the answer to this problem is going to be B 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.

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In Exercises 5–7, perform the indicated operation. Leave answers in polar form. [2(cos 10° + i sin 10°)]⁵