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when we first learn to multiply and divide positive and negative numbers we saw well look if you have a positive number times a positive number I'm just writing pause for short for positive well that of course is going to be equal to a positive number that you saw that even before you learned about negative numbers but it's also true if you had the same sign a negative times a negative that that was also going to be equal to a positive number and the way that you get a negative number is if you multiplied or divided something of different signs so if you have a positive times the negative that would give you a negative number or if you had a negative negative times a positive that also would give you a negative number and here I wrote it for multiplication but it also applies to division a negative divided by a negative is positive negative divided by a positive would be negative now all of these we thought about if we're only multiplying or dividing two things but I'll just think about what happens if we multiply or divide three things or four things or five things or or end things together what we might expect so let's say that we were to multiply a times and now I'm going to write the dot four times a times B times C a times B times C now if I told you that these were all positive numbers then you say okay a times B times C is going to be positive a times B would be positive and then that times C is positive now what would happen if I were to tell you that they were all negative numbers what if a B and C were all negative well if they were all negative let me write it that way let me actually write let's let me write a B and C a B and C they're all going to be negative so if that's the case what is this product going to be equal to well you're going to have a negative here times a negative so a negative times a negative a times B if you do that first and we can when we multiply these numbers that's going to give you a positive so a times B is going to give you a positive but then you're going to multiply that times C you're going to multiply that times C which is a negative so you're gonna have a positive times a negative which is going to be a negative so this one if a times if a B and C are all less than zero then the product ABC is going to be is going to be less than zero as well this whole thing is going to be negative now if I did something else if I said there's other ways that I can make the product negative if a is let's say that a actually let me just write it this way let's say that a is positive B is negative B is negative and C is positive and C is positive well here positive times a negative if you do this first positive times a negative is going to give you a negative and then a negative times a positive different signs is going to give you a negative so this whole thing this whole thing is going to be a negative plus keep on doing we said look if all of them were negative then this thing would be negative that's because I had three numbers you would have had four numbers here what if I had what if I had x what if I had times D here and if I told you all of these numbers were negative let's think about if negative negative negative negative and I can do the multiplication in any order but I'll just go left to right 8 times B negative times a negative that would yield a positive now if you multiply that product times C positive times a negative positive times a negative positive times a negative that would give you a negative and then you multiply this negative times this negative so this whole product ABC is going to be negative but then we multiply times the negative well negative times a negative is going to be a positive so this whole thing is going to be a positive and so you're probably seeing a pattern here if you're multiplying a bunch of numbers if you're multiplying a bunch of numbers and if you have an odd number of negatives odd number of negatives being multiplied and/or divided I just did multiplication here but this would have also been true if these were all division symbols if these were all division symbols we could have we would have been able to say the exact same thing if you have an odd number of negative numbers in your product or in your quotient well then you're going to have a negative you're going to have a negative value for the entire expression if you have an even if you have an even number of negative well then the whole thing is going to be the whole thing is going to be positive and so you can view these as generalizations of what we just saw here positive times a positive you have 0 negative 0 is actually an even number so this would be positive negative times a negative well that's an even number of negatives you have two negatives so that's this case again that's this case again so you're going to be positive either of these cases you have one negative you have one negative in either of these cases so that's going to be this case odd number of negatives so it's going to be a negative and so we can use this knowledge to start dealing with negative numbers and exponents so if I were to say if I were to say I have a to the let me throw a wild number here a to the hundred and first power and we know and we know that a is less than zero what is this going to be well this is taking 101 a's and multiplying them together you have an odd number an odd number of negatives being multiplied together well this whole thing then is going this whole thing is going to be less than zero if we knew what a was we could calculate some out but we know that this thing is going to be negative let's do something so and we could do it other ways we could say something like this if I actually let me try another one what if I tell you that a is less than zero and B is also less than zero and if I had a to the 101 power divided by B to the seventh power what would this expression be would it be positive or negative or zero maybe well you see here a to the hundred first power is you have a you have an odd number of you have an odd number of negative numbers being multiplied so this whole thing is going to be make this whole thing is going to be negative and the same thing applies here you have a negative number to an odd power you have an odd number of this negative number being multiplied together so that's going to be negative but then you have a negative divided by a negative a negative divided by a negative is going to be a positive so this thing right here is going to be positive so this is just a beginning and in the next few videos watch we do a bunch of examples that really test our understanding of this