In Exercises 48–57, perform the indicated operations and write th...

Question

In Exercises 48–57, perform the indicated operations and write the result in standard form. 6/(5+i)

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Hello. Today we're gonna be simplifying a complex number in standard form. So the complex number given to us is 5/4 plus I. Now in order to start simplifying this we're gonna go ahead and multiply by the conjugate of the complex number. The conjugate is just going to be the opposite. So instead of having a positive sign it's going to include a negative sign. So the conjugate is gonna be 4 -1. And we're going to multiply that to both the numerator and denominator. Doing so is going to give me five times -1 Over four plus I Times 4 -1. So let's go ahead and simplify the numerator. The numerator is pretty easy because we just have to distribute five into four minus I. Doing so is going to give me 20 minus five. I. Next in order to simplify the denominator we need to go ahead and foil the denominator, Foiling the denominator we're going to take four and multiply it to four. That's going to give us 16. Next we're gonna multiply 4 to negative. I That's gonna give us -4. I then we're gonna multiply I too positive for That's gonna give us plus four. I and finally we're gonna multiply I two negative I that's then gonna give us negative I squared but what exactly is I squared? Well let's recall that I squared is equal to negative one. That means that we can go ahead and rewrite this foil as 16 minus four. I plus four. I minus negative one. Now let's go ahead and combine like terms together. So negative four I plus four, I is just going to cancel each other out. And negative negative one is going to be positive one And 16 plus positive one is going to give us 17. So our denominator is gonna simplify to 17. Now we need to go ahead and write this complex number in standard form, recall that the standard form of a complex number is a plus or minus B, I plus or minus because this could either be a positive or negative quantity. A represents a regular number and B. Times I represents the complex number. So in order to rewrite this, we're gonna go ahead and separate this into two separate fractions. Doing so is gonna give us 20/ -5/17 I and this is going to be the standard form of the complex fraction. This also means that the solution to this problem is going to be be so I hope this video helps you understand what I square is. And to rewrite complex numbers into standard form and I'll go ahead and see you all in the next video

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