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 am so glad you're here. We are given this expression five divided by three plus I times one minus I were asked to simplify this and write our answer in standard form. What is standard form? Standard form When we have imaginary numbers is where we have A. The constant terms plus B. I. The terms associated with the imaginary numbers. So how do we get it to look like that? Well the first thing we notice is that we have imaginary numbers in our denominator and we can't have that. So we have to get the imaginary numbers out of the denominator. How do we do that? Well what we'll do is when we have our denominator in standard form, when it looks like A plus B. I. We're going to multiply both the numerator and the denominator by the complex conjugate. What is the complex conjugate? The complex conjugate is where you take the A. And the B. I. Terms and those remain the same. And then we flip the sign. So if it was positive it becomes negative. If it was negative it becomes positive. So that will be the complex conjugate but it's not ready for us yet because that denominator is not in standard form yet. There are two A plus B. I's, there's the three plus I in the one minus I. So how do we get rid of those? Well the way to get rid of those is to foil them out. So we'll take three plus I times one minus I. Once we foil that out, it will be in standard form. So the first three times one gives us three Outsides three times negative eyes minus three. I inside I times one is plus one eye and the outside. Or excuse me. The lasts are I times negative I which is negative I squared. Now we can combine our like terms in the middle here so we'll have bring down that 3 -3. I plus I is -2 I. And bring down that minus I squared. We can also simplify that I squared because we remember from previous lessons that by definition I squared equals negative one. So we can replace that here bring down that three minus two I minus and instead of I squared it's minus negative one. We can simplify this minus negative one. Bring down that three minus two I minus negative one is the same as plus one. Now we can combine like terms three plus one. Well that equals four. So we have four minus two. I As our simplified denominator here. So we have five divided by 4 -2. I. And now we're ready for that complex conjugate. So we remember that for the complex conjugate we keep the A term that constant before and we keep the eye term that too I. Or that B. I. And then we flip the sign in the middle. So instead of negative this will be positive and we need to multiply. That was a really long line. We need to multiply both the numerator and the denominator by this complex conjugate by the four plus two. I because four plus two I divided by four plus two. I that's one. We're just multiplying it by one. But it gets the imaginary numbers out of our denominator. Alright, let's go ahead and distribute the numerator Five times four plus two. I five times four is 20 and five times two. I is plus 10. I. Alright we've got the numerator done the denominator that's going to take a few steps because we do need to foil that out. We're going to take four minus two. I times four plus two. I So I'm going to do the steps over on the side here. Firsts four times four is 16. Outside's four times two. Eyes plus eight. I insides negative two I times four is minus eight I and lasts negative two I times positive two. I gives us minus four I squared. Now here's where the complex conjugate is just so cool because look at these inside terms plus eight I minus eight. I. Those cancel out. So we've got 16 minus four I squared and we know That I squared. Well that's negative one. So this is 16 -4 times negative one. We can simplify here. The negative four times negative one is going to be 16 plus four and 16 plus four is 20. So our denominator simplifies to 20. Thank you. Complex conjugate While this is getting close but it's not in standard form yet. We need to simplify this and get our constant term separated from our eye terms. So what we can do is we can factor out in the numerator. We can factor out of 10 both 20 and 10. I are divisible by 10 so 20 divided by 10 is too And plus 10 I divided by 10 is just I. So now we have 10 times two plus I over And the 10 and the 20 can simplify to one and 2. So now we have two plus I divided by two. And now we can take that two in the denominator and apply it to both of the terms in the numerator. So we'll have two divided by two plus I divided by two and the two divided by two simplifies to one plus and then we want the numbers in front of the eye so we'll put one half in front of our eye. And this is our simplified answer. In standard form we look at our answer choices and this matches with answer choice. D. Well done. We'll catch you on the next one

In Exercises 95–99, perform the indicated operations and write the result in standard form.4/(2 + i)(3 - i)

Key Concepts

Complex Numbers

Multiplication of Complex Numbers

Standard Form of Complex Numbers

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