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

5i/(2 - i)

Question

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

5i/(2 - i)

Accepted Answer

Hello everyone in this video we're going to be working on this practice problem that deals with complex numbers, we want to make sure that our answer is in standard form. And the division that we're going to be doing is I divided by eye -2. So you can see that this problem has complex numbers because we have this imaginary components And then we have the real component which in this case is -1. So if I Write that down here, have I divided by I -1. So now I'm gonna rearrange my denominator to be in standard form. And recall that standard form for complex numbers is the real part 1st A plus the imagining part B. I. So if I do that to my denominator, simply rearranging I have I divided by negative one plus I. So now I've rearranged my denominator in standard form. But whenever we're doing division with complex numbers, we want to make sure that the denominator is a real number. And we make it a real number by multiplying by the complex conjugate. So recall that the complex conjugate if my complex number was A plus B. I. The complex contradict would be a minus B. I. So you simply need to flip the sign between the imaginary and real part. So if I multiply by the complex conjugate the complex conjugate of negative one plus I is negative one minus I. And whenever you're multiplying into an expression, you want to make sure that it's the equivalent of one because we're not trying to change the numbers and in this case negative one minus I over negative one minus I is one. So we're not changing the numbers and now we can multiply across. So this becomes i times negative 1 -1. And the denominator is negative one plus I Times -1 -1. So now we can multiply each of these components. So starting with the numerator I times negative one is negative I I times negative I is negative I squared divided by negative one. Times negative one is positive one negative one times negative I is positive I and then positive I times negative one is negative I and positive I times negative I is negative I squared. So here it's important to remember that I squared is equal to negative one Because I is equal to the square root of -1. So using that wherever I see I squared I'm going to replace it with negative one. So my numerator becomes negative I minus negative one divided by one. And you can see that these eyes cancel out. So I'm not going to worry about them. So my denominator is one minus negative one. And now I can simplify this becomes negative by plus one divided by one plus two. One plus one. Sorry. So this becomes negative I plus one divided by two. So now I've simplified it the most I can and I no longer have any imaginary components in the denominator. So if we do, if we separate the real and imaginary components I have negative, I divided by two plus one, divided by two. But remember we want this in standard form where the real component comes first, so the one half has to come first, followed by the imaginary component minus one half. So this becomes my final answer, and this answer aligns with answer choice D. Hopefully this really helped and see you next time.

In Exercises 21–28, divide and express the result in standard form. 5i/(2 - i)

Verified Solution

Key Concepts

Complex Numbers

Division of Complex Numbers

Standard Form of Complex Numbers