In Exercises 21–28, divide and express the result in standard ...

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In Exercises 21–28, divide and express the result in standard form. 2 / 3 - i

Accepted Answer

Welcome back. Everyone in this problem, we want to divide nine by two minus I and we want to write or answer in standard form for our answer choices. A is 18 5th plus nine fifths of IB is 18 5th minus nine fifths I C is nine divided by 41 minus one, divided by 41 I and D is nine divided by 41 plus one divided by 41. I. Now, what are we trying to do here? We're, we're trying to divide a real number by a complex number. And we want to put our answer in standard form and remember that when we say standard form, the standard form of a complex number is A plus B I where A is the real part and B is the imaginary part. So if we are going to write our quotient in standard form, we need to multiply the numerator and the denominator by the complex conjugate of the denominator. Let's break that down a little bit. First, recall that the conjugate of a complex number. So the con complex conjugate equals the same real part, but the opposite imaginary part. So that means for two minus I, it's conjugate would be equal to two plus I, OK. When we multiply a complex number by its conjugate, it turns it into a real number. So that's why we're multiplying the denominator of our, of our co by two plus I its conjugate. But we also have to multiply our numerator by the conjugate because that is how we keep the same value of our expression. In other words, essentially, we're just multiplying by, by, by one. So let's go ahead and do that. And then we'll see if we can write it in standard form. No, nine, multiplied by two is 18 and nine multiplied by I is nine. I in our denominator. One of the benefits of using the complex conjugate is that it brings out a very important property. Now look at both of them, two minus I and two plus I. Do you notice anything here? Well, wouldn't you recall that it's representing the difference of two squares? Because remember the difference of two squares tells us that if we're multiplying the sum and difference of two expressions, it's the same thing as writing them as the difference of both their squares. So A plus B multiplied by A minus B equals A squared minus B squared. We can use that idea here because we can rewrite two minus I multiplied by two plus I using that property as two squared plus I squared or minus, sorry, two squared minus I squared. No one of the reasons why this is good is because I squared is a real number. Again, recall that given I equals the square root of negative one, then I squared would be the square root of negative one squared which equals negative one. So that means I can rewrite. My expression is 18 plus nine, I all divided by two squared, which is four minus I squared, which is negative one, no four minus negative one is four plus one which equals five further. I can separate the values in my numerator. So that would have been 18 divided by five plus nine, divided by five. I no, my expression is in standard form where it has a real part and an imaginary part. And when we look back on our answer choices, 18 5th plus nine fifths of I would have been answer choice. A thanks a lot for watching everyone. I hope this video helped.

In Exercises 21–28, divide and express the result in standard form. 2 / 3 - i

Key Concepts

Complex Number Standard Form

Division of Complex Numbers

Complex Conjugate

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