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

Question

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

- 6i/(3 + 2i)

Accepted Answer

Hello today we're going to be rewriting a complex expression in standard form. So the complex expression given to us is negative two I over the quantity one minus I. Now in its current form there's really nothing much that we can do to simplify this further. However notice that in the denominator we have a quantity that involves an imaginary number whenever you have that you can go ahead and multiply by the conjugate of the imaginary number. That way we may be able to rewrite this in standard form. Now the conjugate of the denominator is the same quantity but with an opposite sign. So the conjugate is going to be one plus I over one plus I. So we're going to be multiplying the numerator and denominator by this new quantity. Doing so is going to give us negative two I times the quantity one plus I over one minus I times the quantity one plus I Now in order to simplify the numerator, we're gonna go ahead and distribute negative two I 21 plus I. So doing that is going to give us negative two I plus negative two I times I which is going to give us negative two I squared. And in order to simplify the denominator we're going to need to foil those quantities together. So we're going to need a foil, one minus I and one plus I. So foiling this is going to give us one times one which is just one plus one times I which is going to be I negative I times one which is going to be negative I and negative I times positive I which is going to give us negative I squared now I minus I is just going to cancel itself out. So what we are left with is one minus I squared but keep in mind that I squared is equal to negative one. So you can go ahead and replace I squared with negative one. Doing so is going to give you one minus negative one which is the same as one plus one, which results in two. So our denominator simplifies to just to now further simplifying the numerator again recall that I squared is equal to negative one so we can go ahead and replace this I squared with negative one. Doing so is going to give us negative two. I plus negative two times negative one. And that entire quantity is going to be over two. Now negative two times negative one is going to give us positive two. So what we have is negative two. I plus 2/2. Now in the numerator, both of these terms have a factor of two so we can go ahead and factor out a two in the numerator. Doing so is going to give us two times negative I plus 1/2. And since we have two in both the numerator and denominator we can go ahead and cancel those quantities out what that's going to leave us with is negative I plus one but I'm gonna go ahead and rewrite this with the positive term in front. So what we are left with is 1 -1. And this is going to be the standard form of this complex expression. That also means that the answer to this problem is going to be be So. I hope this video helps you in understanding how to simplify a complex expression, and I'll go ahead and see you all in the next video.

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