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.

2/(3 - i)

Accepted Answer

Hello. Everyone in this video we're going to be working with complex numbers and we want to make sure that our answer is expressed in standard form and we're going to be performing the Division 24 divided by 1 -1. So in this case you can see that we have the imaginary component in the denominator with the I present here. So when we perform a division we want to make sure that the denominator has only real parts. So to get the denominator to have only real parts, we multiply by the complex conjugate. So the complex conjugate of a complex number. Say we have A plus B. I. Than the complex conjugate is a minus B. I. So notice how the plus went to minus. So we're going to do that same thing for our complex number which in this case is one minus I. But We need to do one plus I remember that we want to do one plus I over one plus I because we don't want to add any extra components and if we do one plus I over one plus I it's the same thing as multiplying by one. So if we do this multiplication, we're left with 24 times one plus I on the numerator and on the denominator we have one minus I. Times one plus I. So we can go ahead and carry out this multiplication. So 24 times one is 24, times I is 24. I That's being divided by one times one. So that's one one times I which is I negative I times one is negative I negative I times positive I is negative I squared. Notice how the two eyes in the denominator cancel out. So plus I minus I. This is the same thing as zero So I can rewrite this as 24 plus 24. I divided by one minus I squared And in this case recall that I is equal to the square root of -1. So if I have I squared that is equal to negative one. So wherever I have I squared I'm going to write -1. So I have 24 plus 24 I divided by one minus negative one replacing the I squared with negative one. So my denominator becomes one plus 21 plus one. So to so now I have a denominator that only has real components. So I'm gonna split up my numerator so that I can have an imaginary component and a real component. So I have 24 divided by 2-plus 24 I divided by two. So now I have a real component and an imaginary one. And if I simplify these fractions I get 24 divided by two is 12 plus 12. I and this is already in standard form because I have the real component first and then the imaginary. So this becomes my final answer and you can see that this aligns with answer choice D. So hopefully this video helped and see you next time

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