Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Complex numbers are in the form A+, BI where A&B are both real numbers. Real numbers being square roots, cube roots, integers, fractions, decimals, etcetera. Now when we combine complex numbers, we're going to add like terms. So anything that's the a portion or the real portion is going to get added together and anything that's the imaginary portion or the portion multiplied by the I is going to be able to be added or subtracted together. The definition of I is the square root of a -1. So if we thought about I equaling sqrt -1, then I squared square. Each side would equal -1. That's what makes it a complex number. Imaginary equals sqrt -1. So if we look at this first example sqrt -36, we could think of that as sqrt -1 * 36. The square root of the -1 is I sqrt 36 is 6. Traditionally we write the I at the end, so sqrt -36 is the number six I sqrt -13 that's -1 * 13, sqrt -1 is I sqrt 13. We don't know. We frequently write the I at the end, but make sure it's very, very careful that you don't have that I underneath the square root. Sometimes, because of the confusion, we do write the I in front. Both of these are acceptable. Negative sqrt -25. Well, that negative inside the root is going to come out as an I. The 25 sqrt 25 is 5 S Our final answer is going to be -5 I #4 sqrt -9 negative one times sqrt -27 negative one times. If we thought about prime factoring 27 three times nine 3 * 3 * 3 so 3 ^3. The understood index of two goes into the three one full time with one leftover and the square root of the -1 is I. So our final answer, 3 root 3I or sometimes we put the I in front so that it's not confused with being underneath the radical of three I root 3. Both forms are OK sqrt -75. We're going to think of -1 * 75. If we thought about prime factoring 75, we'd get 5 * 15 or 5 * 3 * 5, so sqrt -1 * 3 * 5 ^2. The square root of the negative is I sqrt 3, we don't know, and sqrt 5 ^2 the square root and the square cancel, so we'd get 5I root 3 or sometimes written as five root 3I. We can add and subtract if they're like terms. So we're going to figure out prime factor 76 two times 3838 is 2 * 19. We're going to prime factor 125 five times 25 or 5 * 5 * 5. So we have negative square root -1 * 2 ^2 * 19 + -1 * 5 ^3. Well, the negative in front stays. The square root of a -1 is I sqrt 2 ^2. The index of two goes into the exponent of two one full time with none leftover. So the Two's going to come out. We don't know sqrt 19. The second term here sqrt -1 is I. The index of two goes into the exponent of three one time with one leftover so -2 sqrt 19 + 5 sqrt 5. That whole thing is being multiplied by the I imaginary number. Is the form A plus BI where B is equal to or not equal to 0? A complex number is a plus BI where A or B could be zero and a real number is the form a plus BI where B does equal 0A and B are real numbers. So examples of complex numbers 5 + 7 I -7 -, 3.6 i7 I 16 zero. The real portion on the first one is 5, the imaginary is 7, The real on the next one is -7. The imaginary is -3.6. The real on the next one was 0, the imaginary is 7, the real on the next one is 16 and the imaginary is 0. The real on the last one is 0 and imaginary is also zero. So complex numbers made-up of A+, BI, where A&B are both real numbers, but the real number B is being multiplied by the I, which makes it the imaginary portion of the complex number. We're going to be able to add and subtract and simplify. So when we look at this, we're going to add the real number portions of 5 + 7 and we're going to add the real number or the imaginary portions of the i's. So 5 + 7 is 12 two I -, I is going to be a single I 4 + -13 and -5 I +2 I. So 4 + -13 is -9 negative 5I plus two I is -3 I. Literally, we just combine our like terms, the real portions, we combine the imaginary portions, we combine this next one. We need to distribute the negative first. So we get -2 - 4 I 9 - 2 is 7 seven I - 4 I IS3I. Once again, we're going to distribute the negative, so we get +7 - 5 I -5 + 7 is 2 negative I - 5 I is -6 I. We're combining the real number portions and the imaginary number portions. Sqrt 16 is 4. The square root of negative is that I and 25 would be 5 S Our first parentheses is going to be -4 -, 5 I plus 22 -, sqrt a negative is I sqrt 9 is 3, so three I. When we combine our like terms -4 + 22 is 18 -5 I -3 I is -8 I. When we multiply, we're going to multiply just like we would anything else. So we're going to have the seven times, the six to give me 42 I times I is I ^2. Now, if we remember back to our definition, if I is squared and -1, if we square one side, we square the other, so we know I squared is -1. So we're going to have 42 * -1 or -42 when we have seven I * 6 I here sqrt -36 sqrt. A negative is the I sqrt 36 is 6 * sqrt. A negative is I sqrt 9 is 3, so six I * 3 I IS18I squared. Remember I ^2 is really -1 because our definition said I is sqrt -1, so we'd have 18 * -1 or -18 squared and -5 Times Square to -2 sqrt. A negative is I sqrt 5. We don't know, so we're going to leave it. Square root of a negative is I again sqrt 2. We don't know, so we're going to leave it underneath the root I * I is I ^2 sqrt 5 times sqrt 2 is sqrt 10. I squared is -1 so -1 times root 10 or negative sqrt 10 -6 times sqrt -6 times sqrt -21 sqrt -6. We could think of as -1 * 6, sqrt -21 negative 1 * 21, so I sqrt 6. I'm going to think about prime factoring that six is 2 * 3, sqrt -1 I times 21, three times 7, so I get I ^2. When I multiply inside the radicals, I'm going to have 2 * 3 ^2 * 7. The I ^2 is going to be -1. We don't know sqrt 2, but sqrt 3 ^2, a square root and a square are going to cancel, leaving us a three sqrt 7. We don't know. So our final answer is going to be -3 square roots of 14. Before using the product rule for radicals, you must convert in terms of I. So in this first one, they're already in terms of I. So we're going to do distributive property 5I times two is 10 i5 I * 7 I IS35I squared I ^2, however, is -1. So to put it in a plus BI form, we're going to have -35 + 10 I. We're going to distribute again. So we're going to get -21 I plus 28I squared I ^2 is just -1 so -21 I plus 28 * -1 negative 28 - 21 I when we have it in a plus BI form. Now we're going to have to foil. So first outer inner last 6 * 318 six times 4I 24 I -5 I times three -15 I and -5 I times 4 I -20 I squared. Remember, if we put our lines like I've just shown it and we squint and we have a bit of an imagination that looks like Charlie Brown, we need to simplify this up. This I ^2 at the end is really going to be a -1. So we're going to have 18 + 24 I -15 I. And instead of a -20 I squared I ^2 is -1. So the opposite of -20 would be positive. So our final answer is going to be 38, adding the 18 and the 20 + 9 I, the 24 I -15 I that's in our A+ BI form. This next one, 3 - 2 I really means 3 - 3 - 2 I. Quantity squared means 3 - 2 I times 3 - 2 I. So we need to foil this one again. Our first terms together are going to give me the 9. My outer terms are going to give me -6 I, my inner terms, another -6 I. And my last terms positive 4I squared, that I squared is really a -1. So we're going to have 9 - 6 I -6 I minus four, 9 - 4 gives us 5, and -6 I and -6 I is -12 I. We're going to now rationalize. We're going to use the concept of a conjugate. The conjugate means keep everything the same except for changing the sign in the middle. So if I have 3 + I, the conjugate is 3 -, I. If I do the bottom, we need to do the top. The whole reason of using the conjugate is based on the difference of squares, and what will happen is the i's will all cancel when I foil the bottom. I'm going to get 9 -, 3 I plus three I -, I ^2. But that I squared is really just -1. So the 9 -, 3 I plus three I -, A negative one makes it a plus. So the -3 I and +3 I will cancel, leaving me 10 in the bottom. Hence all the imaginary numbers have gone away in the top. We actually don't want to distribute till the very last step. And the reason is that we may have things that cancel, and if we haven't distributed, it's a monomial. So 4 - 3 I over 10. This four and 10 are each divisible by a two. So we're going to get 2 * 3 -, I / 5. Now when we distribute, we get 6 - 2 I over five. It wants our answer in a plus BI form. So we're going to have 6 fifths minus 2/5 * I. The next one. The conjugate here is going to be 5 + 3 I. If I do the bottom, we have to do the top. I'm not going to distribute out the top until the final steps in case I can get things to cancel. When I foil out this bottom, we get 25 + 15 I -15 I -9 I squared. Well, that -9 I squared at the end is really going to be +9 and the +15 I and -15 I'll cancel. So we get 25 + 9. 25 + 9 is 34. I'm leaving that numerator alone. Now we've got a monomial over a monomial, and this four and that 34 are each divisible by two. So we get two I * 5 + 3 I over 17. Now we're going to distribute, so we get 10I plus six I squared. That I squared remember is really just a -1. So ten a -, 6 / 17. Putting it in a plus BI form, we get -6 seventeenths plus 10 seventeenths I when there's only one term we can actually get by with just multiplying by an I. And the reason is that if we have nine I squared on the bottom, we know that I squared is really just -1 So our bottom would turn into -9 our top. If we go ahead and distribute, we'd get 3I plus 8I squared three I -, 8. We don't leave a negative in the bottom, so we're going to multiply by -1 / -1. This is probably the easiest way to think about it. So we get -3 I +8 over +9. If we want to give our answer an A+ BI form, we'd have eight ninths -3 ninths I. Oh, but this three ninths can reduce, can't it? So we'd get 8 ninths -1 third I as our final answer. Now you might wonder why I went ahead and multiplied the I back here, whereas in the past I don't multiply the numerator. It's because there wasn't a coefficient that would cancel, and I know that in the denominator we're only going to have real numbers. Once I get rid of the imaginaries, this next one we're going to multiply by the conjugate 7 + 4 I 7 + 4 I. This time I am going to multiply out because I don't have a real number in the numerator that could cancel. So when I foil, we're going to get 35 + 20 I plus 21I plus 12 I squared. When I foil out the denominator, we're going to get 49 + 28 I minus 28I. -16 I squared, so 35 + 20 I plus 21 I. That 12 I squared is going to turn into -12 on the bottom 49, the +28 and -28 are going to cancel. The -16 I squared is going to turn into a +16. So now if we combine like terms, 35 -. 12 is 2320 I plus 21 I is 41 I. 49 + 16 is 65 S Putting it in the appropriate form. 2360 fifths plus 4160 fifths I A+ BI 2360 fifths is a real #4160 fifths is a real number being multiplied by the imaginary I here 3 - 6 I on the bottom, so 3 - 6 I on the top. When we foil 15 -, 30 I -6 I plus 12 I squared all over 9 - 18 I plus eighteen I -, 36 I squared, so 15 - 30 I minus six I -. 12 / 9 the -18 I and +18 I cancel so -36 I squared turns into a + 36. 15 - 12 is three negative thirty I -. 6 I is -36 I. 9 + 36 is going to give us 45, so putting it in the appropriate form. 340 fifths -3640 fifths I, but 340 fifths have a three in common, so a three divides into three once, and a three goes into 4515 * 36 and 45 have a nine in common. 36 / 9 is 445 / 9 is 5 S 115th -4 fifths I I to the 11th. Well if I = sqrt -1 and I ^2 equal -1, what would I cubed equal? Well, I cubed we could think of as I * I ^2 and I squared was really -1. So I cubed would be I * -1 or negative I. I to the 4th we could think of as I ^2 * I ^2. I ^2 is really -1 and -1 * -1 is a +1 I to the fifth. Well, I to the 4th is 1. So if we thought about it as I to the fourth times I, we'd have one times I or I. Now there's a pattern that's going to develop. We thought about I to the 6th that could be thought of as I to the fourth times I ^2. So 1 * -1 or -1 I to the 7th we could think of as I to the fourth times I ^3. I to the 4th was one and I cubed was just negative I so we'd get negative I. I to the 8th we could think of as I to the fourth times I to the 4th or 1 * 1, which is one I to the 9th we could think of as I to the eighth times I I to the 8th was one times I would be I. If we look here, what's really happening is every multiple of four is going to be one every term. That's one less than a multiple of four. IE I cubed is 1 less than four is negative I, I to the 7th is 1 less than eight. It's negative I if we thought about one more than a multiple of 4, I to the fifth is I, I to the 9th is I. And if we went way back here, I to the zero, anything to the zero is 1. And that technically is a multiple of four. So there's my one. So one more than that multiple of four would be I to the first, which is I. If we thought about two less than a multiple of four or two more than a multiple of four, they're the same thing. We'd get I ^2 as -1 I to the 6th is -1 I to the 10th would have been -1 etcetera. So with all that explanation, the easiest way I can tell you to do these is to think about 11 is really a multiple of 4/8 with three leftover. Well, any multiple of four is 1 and I cubed is negative I. So our answer here is going to be negative I. So think about 11 / 4 or how many times four goes into 11, that would be twice or eight times total with three leftover. So I to the 8th was one I cubed is negative I, I to the 37th. Well, 36 is a multiple of 4. There'd be one leftover to get us to the 37. I to the 36. Any multiple of 4I to a multiple of four is always one. So I to the 37th is just going to be I negative I to the 71. You need to think of this as -1 to the 71 times I to the 71. Well, a -1 to an odd power is going to give us a -171 we could think of as 68, which is a multiple of four with three more. So -1 I to the 68 is just one, and I cubed is negative I. So if I have -1 * 1 * -I that's really going to all simplify down to I negative I to the 4th. Think of it as -1 to the 4th times. I to the 4th -1 to an even power is going to give us one. I to the 4th, that's a multiple of four. So it's one. So we're going to get 1 * 1 or one -3 I to the 5th -3 to the 5th, I to the 5th -3 to the 5th is going to be -243 I to the 5th. We could think of as I to the fourth times I. Any I to a power of four is really one. So 243 * 1 * I or -243 I our last 15I to the 5th 5I to the fourth times I + 4 I cubed. Well that I to the 4th is one the I cubed is negative I so we get five I -, 4 I or just I.