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.

8i/(4 - 3i)

Accepted Answer

Hello. Everyone in this video we're going to be looking at this practice problem where we were dealing with complex numbers and we want to make sure that our answer is expressed in standard form. After we performed the division seven I divided by three I plus five. So you can see that we're dealing with complex numbers because we have this imaginary component and then the real component. And whenever we're dividing with complex numbers it's important to remember that we want to get rid of the imaginary components in the denominator. So you can see that we have an imaginary component here in the denominator with the three I. And the easiest way to get rid of the imaginary components in the denominator is to multiply by the complex conjugate of that complex number. So recall that if we have a complex number such as A plus B. I. The complex conjugate is a minus B. I. So the sign switches between the imaginary parts and the real parts. So if I rewrite my denominator in standard form it becomes seven I divided by five plus three. I. So now I can multiply by the complex conjugate which is 5 -3 I. And recall that I want to make sure to not add any numbers to my Division. So if I multiply by 5 -3 i times 5 -3. I it's the same as multiplying by one. So I can perform this division and hopefully get rid of that imaginary component in my denominator. So my numerator is seven I times 5 35 I seven I times negative three I negative 21 I squared. My denominator is five times 5 25. 5 times negative three. I negative 15 I positive three I times five positive 15 I and then positive three I times negative three I negative nine I squared. So you can see in my denominator that these middle terms negative I and positive 15. I cancel out. So I can say that those are just zero and I'm left with 35 I minus 21 I squared Divided by minus nine I squared. Well multiplying by the complex conjugate. I'm still left with this I component or imaginary component in my denominator. So it seems like multiplying by the complex conjugate didn't work. But in this case recall that I is equal to the square root of negative one. So if I do I squared I have the square root of negative one times the square root of negative one. And when you have two square roots multiplied together You're left with just what's inside. So notice how -1 is a real number. So now I'm going to go back to my equation and wherever I see I squared, I'm going to replace it With -1. So my numerator becomes 35 I -21 times negative one. And my denominator becomes 25 - times negative one. And if I do that multiplication, I'm left with 35 I negative 21 times negative one is positive 21 And my denominator is and then we have negative nine times negative one. So positive nine And I'm left with 35 I plus 21 Divided by 34. So you can see that at first glance it seems like the complex conjugate did not get rid of my imaginary components, but once I substituted in that negative one my denominator is now only made of real numbers or number in this case. And now I'm going to separate my fraction because you can see that in my numerator I still have the imaginary components and the real. So I'm gonna separate those two. So I'm left with 35 I divided by Plus 21 divided by 34. So because these fractions can't simplify to anything, this could be my final answer. But remember we want it in standard form and standard form. The real parts are written first So I'm just going to switch the order. So I'm off with 21 divided by 34 Plus 35. I divided by 34 and this becomes my final answer and you can see that this answer aligns with answer choice A So hopefully this video helped and see you next time

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