Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College Complex numbers. The definition of the imaginary number I is equal to sqrt -1. Thus, if we square each side, we can see I squared equal -1 imaginary numbers in the form of a plus BI where A equal 0 complex numbers A+, BI where A&B are both just real numbers and a real number is a plus BI where b = 0. So a complex number is made-up of an imaginary part and a real number part. Now with the complex number A or B could be zero or they both might be 0. So examples of complex numbers 5 + 7 I -7 -, 3.6 i7 I 16 zero. A real number can be positive or negative. It can be a whole number of fraction, a decimal, 0, etcetera. So our first problems are going to say express in terms of I sqrt -36. We could think of a sqrt -1 * 36. Well square to -1 by definition is I sqrt 36 is 6 and we usually put the I at the end, so six I. This next one we can think of as -1 * 13 sqrt -1 we know is I sqrt 13. We don't know. So sqrt 13 stays on side sqrt 13 * I this next 176. So we're going to get sqrt 76. I 76 is divisible by 238. That's divisible by two. So actually square to 76 we could think of as negative -1 * 4 * 19. So in this case we can see this is going to be negative sqrt -1 was I by definition sqrt 4 is 2 and the 19 we don't know so we're going to leave it inside 125. If we prime factor it we get 5 * 25 five times 5 * 5. So we can think of this as -1 * 5 ^3, and the index of two goes into the exponent of three one whole time. So we're going to take out 1/5 with 15 leftover. The leftover stay on the inside. So that's going to be our final answer. You're going to have a few that you're going to try. We're going to perform the indicated operation. So we're just going to add and subtract and then simplify. So -5 -, I minus the negative turns into plus minus a positive. Using distributive property turns into a negative. So -5 + 7 is 2. We're going to combine our like terms, so we get 2 -, 6 I. This next one we're going to get -4 -, 5 I plus 22 -, 3 I. So just combining -4 + 22 is 18 negative five I -, 3 I -8 I. This next one we're going to get -3 - 6 I -14 - 2 I. So once again, we got to remember this distribution of the negative so -3 -, 14 negative 7 negative six I -. A negative 2 -. A negative makes it plus so -4. I got a couple for you to try. So we're going to perform the indicated operation by multiplying or dividing here. Note, before using the product row for radicals, you must convert the terms to I. So seven I * 6 I is going to be 42 I squared, but I squared is -1 so 42 * -1 or -42. This next one, we're going to get sqrt 5 * I sqrt 2 * I. So sqrt 10 I squared, but I squared was -1 so negative square roots of 10. We've got some for you to try. And then we're going to actually, I think maybe I'll do this one here. Let's foil this one out. So first outer, inner last. So 18 plus 24 I -15 I -20 I squared. So we'd get 18 positive, 24 negative 15 is going to give us +9 I. This I ^2 at the end is -1 though, so -20 * -1 is going to be a +20. So we're going to get 38 + 9 I. So the next type we're going to do is perform the indicated operation, divide complex numbers and simplify right each insert in the form a plus BI. If we've got a single I in the denominator, we're going to multiply by I / I. And what that'll do is that'll give us a nine I squared in the bottom, and nine I squared is really going to be -9 because I squared is -1 And then we're going to pull the negative out in front and multiply it through by the top. So if I multiply the bottom by I, I've got to multiply the top by I and I'm going to get 3I plus eight I ^2 I ^2 once again is really just -1 so we get three I -, 8. So when I have this negative in the top, I'm going to distribute that negative through so -3 I plus 8 / 9. Now the directions say to write it in a plus BI form. So my real number portion is 8 ninths plus -3 ninths I. But we know that -3 ninths is really one third. So eight ninths plus negative 1/3 I. This next type we have to do by what's called a conjugate. And the conjugate is leaving everything exactly the same but changing the sign in the middle. And it's based on the difference of two squares. When I multiply this out, I'm going to see that we're going to end up having the eyes all cancel the denominator. So we're going to get 49 + 28 I -20 eight I -. 16 I squared. What just happened? OK, we're going to try that again. I'm guessing I'm riding too close to the edge. So 35 + 20 I plus 21I plus 12 I squared on the top and 49 + 28 I -20 eight I -, 16 I squared on the bottom. So the -28 I and positive 28I on the bottom are going to cancel. All the I squares are going to turn to -1. So we're going to end up with 35 + 41 I -12 on the top over 49 + 16 on the bottom. So 35 -, 12 is going to give us 23 + 41 I over 65. It wants it in a plus BI form. So 2360 fifths plus 4160 fifths I. And then there's a few for you to try. The next type we're going to have is we need to simplify I to a power. And there's actually a pattern that's going to occur here. If I know that I squared is -1, then if I thought about I cubed, I could think about multiplying each side by I. I here and an I here. So I cubed is really the same thing as negative I. If I thought about I to the 4th, I could take I cubed and multiply each side by I, so that would end up being negative I ^2. But I squared is -1 so the opposite of -1 is +1. So what's going to happen is we're going to have a pattern that's going to go I, I ^2 is -1 I cubed is negative I. I to the 4th is one I to the fifth is going to come back around to being just I because if I multiplied each side by I. So this one's going to be I to the 6th, I to the 7th, I to the 8th. So it's going to repeat every multiple of 4. We want to write our answer in terms of i's or negative or positive ones or i's. So when I look at this next example, I'm going to have negative I to the 71st. So I'm going to think of -1 to the 71st times, I to the 71st, well -1 to the 71st is -1 and I to the 71st. I want to think about multiples of four because I to the 4th is one. SO4 goes into 7117 times because that's 68 with three leftover 68697071. So this is going to be 1 to the 17th. And we know one to any power is one and I cubed is just negative I. So this whole thing is going to simplify to positive I because a negative times a negative is a +1 to the 17th is 1. So we get just plain I in the end, negative I to the 4th. The next example -1 to the 4th times I to the 4th, well -1 to the 4th is one, and then I to the 4th is going to be just one. So that one's 15I to the 5th, SO5I to the fifth is going to be 5I to the fourth times. One more. I remember we add exponents when we're multiplying the bases plus 4I cubed. Well, I cubed was negative I. So we're going to get five I -, 4 I or a single I when we're done and a couple for you to try. Thank you and have a wonderful day.