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

Question

In Exercises 21–28, divide and express the result in standard form. 2i/(1 + i)

Accepted Answer

Hello. Everyone in this video we're going to be looking at this practice problem where we're trying to perform the division For I divided by three plus I. So in this case you can see that we're going to be dividing with complex numbers. And whenever we're dividing with complex numbers, we want to make sure that there's no imaginary components in the denominator. And in this case you can see that we have an imaginary component with the I. In the denominator. So in order to eliminate that, we're going to use the complex conjugate. And an easy way to think of the complex conjugate is to just switch the signs between the real term and the imaginary term. So in this case our real term is three and our imaginary term is I. And after I switch the signs between those two, I'll be left with minus. So I have three minus I. Instead of three plus I. And we want to make sure to multiply by one. So three minus I divided by three minus size the same as one. So we can complete that multiplication. Now we have four I times three and the numerator is leaving us with I and for I times negative I is negative for I squared and then we can do the same to the denominator. We have three times three is 93 times negative I is negative three, I I times three is positive three I and I times negative I is negative I squared. So now you can see that the middle terms in the denominator negative three I and positive three. I canceled 20. So we're actually left with 12 I minus four. I squared divided by nine minus I squirt. And in this case recall that we multiplied by the complex conjugate to get rid of the imaginary components. But we still have I squared. Well recall that I is equal to the square root of -1. So if we do I squared that is the same as I times I which is equal to the square root of negative one times the square root of negative one. And when we multiply the square root of negative one times the square root of negative one. We're left with a real number negative one. So I squared is really a real number. So now I can go back to my equation And wherever I see I squirt, I'm going to plug in -1. And if I do that I'm left with 12 I - And I plug in negative one Divided by 9 -3. So if I simplify that I have 12, I negative four times negative one is positive for divided by nine, Two negatives turn positive. So nine plus one. So my final Equation will be 12. I plus four divided by 10. And now you can see that this final equation doesn't have any complex or imaginary numbers in the denominator. So I can now complete my division and I can complete the division by separating this fraction into real components and imaginary components. So I can have my imaginary components 12. I divided by 10 plus four divided by 10. And from here you can see that there's some simplifications so you can divide 12 divided by 10 is the same as six divided by five And four divided by 10 is the same as two divided by five. So this becomes the final simplification and we can rewrite this to have the real number in front two divided by five plus six, I divided by five and we want to rewrite it with the real number in front because the question asked us to write in standard form and in standard form the real component comes first so this becomes our final answer and you can see that this aligns with answer choice C. So hopefully this video helped and see you next time.

In Exercises 21–28, divide and express the result in standard form. 2i/(1 + i)

Key Concepts

Complex Numbers and Standard Form

Division of Complex Numbers

Complex Conjugate

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