In Exercises 95–99, perform the indicated operations and write th...

Question

In Exercises 95–99, perform the indicated operations and write the result in standard form.

4/(2 + i)(3 - i)

Accepted Answer

Welcome back I'm so glad you're here. We have got this expression we've got five divided by three plus I times one minus I all in the denominator and we are to simplify and then writer answer in standard form and what is standard form when we're dealing with complex numbers it's got an eye in it. That is going to be A plus B. I. So how do we get this looking like standard form while one thing with complex numbers is that we do not want complex numbers in our denominators and we want to work to get those out and to get that out. We're going to multiply it by its complex conjugate when you have got a number in standard form. Its complex conjugate as we recall from previous lessons is going to keep the A. And it's going to keep the B. And the I. Values and flip the sign. Great. Except we have two complex numbers in our denominator. We have this three plus I times one minus I. So we're going to have to go ahead and foil that out so that we can have it looking like a single complex number and then multiply it by its complex conjugate. So let's go ahead. We keep the five on top. And to foil we've got three times one for our first is three. Our Outsides three times negative I is minus three. I our insides I times one is plus I and our lasts I times negative I is minus I squared. Alright we have some like terms here in the middle and we want to recall from previous lessons that by definition I squared equals negative one. So we can replace that I squared with a negative one. This will simplify as five divided by three then negative three I plus I is minus two I And -20. We can combine these like terms here we've got R. Three and r minus negative one which is plus one. So we've got five divided by three plus one is four minus two. I and hey look at that that four minus two. I does look like a complex number in its standard form and it looks more like this red one, it looks like it has a negative in it. So we've got four minus two I. And its complex conjugate is going to keep the four that a value keep the to the B. Value in the eye and flip the sign. So we're going to multiply this by four plus two. I. And what we do to the denominator we also have to do to the numerator because anything over itself is just going to be one. So it's like we're multiplying it by one for the numerator. We're going to distribute that 55 times four is 20 and five times two I is going to be 10 I in the denominator. We could go ahead and foil that out again. But as we recall from previous lessons when you multiply a number by its complex conjugate. You're going to get a squared plus B squared. So in this case A. Is four. So we're going to have four squared and B is two so plus two squared K four squared is 16 plus scoops not to the fourth. Two squared Two squared is four, so 16-plus 4 is 20. And that is now our denominator. This is very close to being ready but we want to have it in standard form where we separate out our eye numbers from our non I numbers. So we've got 20/20 plus 10/20 I and 20/20 is one plus 10/20 is one half I. And that is as simplified as it's going to get we look at our answer choices and that matches with answer choice. D. Well done. We'll catch on the next one.

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