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what I hope to do in this video is a bunch of examples to show us that a lot of the properties we've been dealing with in arithmetic the distributive property associative property commutative property that these apply just as well to negative numbers but with that said it is good to actually see it used using using negative numbers or see these properties applied just to make sure that we we understand what's going on so these exercises these are all from Khan Academy whoops so this first one says which of the following expressions are equivalent to negative 2 times the quantity 5 minus 3 now you could of course just figure out what that 5 minus 3 is 2 and then multiply that times negative 2 and you would get negative 4 and you can see which of these is equal to negative 4 and that would be fair but the whole point of this video is to understand that look maybe I could apply the distributive property here so let's do that so what I could do is I could distribute this negative 2 I can multiply it times 5 and then I could multiply it by I could either view it as it's going to be negative 2 times 5 plus negative 2 times negative 3 or you could view it as negative 2 times 5 minus negative 2 times positive 3 now let me write those two things down so you could view this as negative 2 times 5 negative 2 times 5 plus negative plus negative 2 times negative 3 you could view it that way or you could view it as negative 2 negative 2 times 5 minus minus and if I'm putting a minus here that I'm going to view this as a positive 3 that we're subtracting a positive 3 so minus negative 2 times positive 3 notice I either wrote the negative here and we're positive here I wrote the negative here and made this a positive 3 but these are going to be equivalent either way I've distributed this negative 2 notice I have a negative 2 negative 2 and what are these going to be equal to well negative 2 times 5 is negative 10 and then and then negative 2 times negative 3 is positive 6 or over here negative 2 times 3 is negative 6 but then we tract it so just going to get positive you're going to get positive six either way this right over here is positive six and this over here subtracting a negative 6 would give you positive six so you get negative 10 plus 6 and that's this choice right over here which of course does evaluate to negative 4 which would which this expression does evaluate to this wood up here evaluates to negative 16 and of course I won't select this because I found an answer let's do of several more which of the following expressions are equivalent to negative s times T times s select all that apply and here we can't just substitute we can't just evaluate it and see what is this these evaluate to we should we used to do a little bit of a manipulation of these variables well there's a couple of ways to think about it one we could change the order in which we multiply these things so we could view this as negative s that's that let me write it a little bit neater we could view it as negative s times s times s times T times whoops let me do that a different color times s times T times T and do any of the choices look like that well almost this instead of saying negative s times s it says s times negative s and because multiplication once again you know I I'm not a big fan of using the word because it sounds complicated but just it's it's commutative a times B is the same thing as B times a so I can rewrite this as I can rewrite this I can swap these two and write this as s times s times negative s times T times T times T all I did is I swapped these two this negative s and this s I just swapped them and I got exactly what I have right over here now let's just make sure that this one does not apply maybe the easiest ways to try to simplify this and the best way I could think about that is by distributing this s so if I distribute this s what I'm going to get this is going to be equal to s times T which is s T or I could even I can write it like this I could write it s times T like that and then I have minus s times s so minus s times s I could write it that way or I could write minus s squared if I want to that's the same thing as s times s but this is very different this is very different here I'm just taking the product of three variables here I have two different terms taking the product of two variables here and then the product I guess you could say I'm taking s squared I'm picking s times s so this is not this is not the same thing which of the following expressions are equal it to negative x times and then in parentheses negative y times X and I forgot to mention it but like always pause the video try to work them out by yourself before I do them all right select all that apply so let's just try to manipulate this a little bit so once again multiplication it's associative I could I could so it's negative X x times negative Y times X so the way it's written here I could do these first that's essentially what's written over here or its associated I instead I could do these first and the reason why I find this interesting is a negative times a negative is going to be a positive so this is going to be the same thing this thing over here is going to be the same thing as positive x times positive Y negative times a negative is a positive so we're going to get positive x times y and then you're multiplying by an X again multiplying by an X again now the other thing we know about multiplication is it's commutative we can change the order in which we multiply because I don't see I don't see this quite I don't see this going on over here yet so let's see if I change the order if I put the X's if I multiply the X's first I could write this as I could write this as x times X x times X times y times y all I did is I swapped these two once again that can swap the order when I'm multiplying and I don't quite see this yet but x times X that's the same thing as X square this thing right over that's x squared so this is going to be x squared x squared times y which is exactly what we have over there now what does this one up evaluate to and this one actually evaluates to a number because regardless of what X you pick X minus X that's going to be 0 0 times anything is going to be 0 so this thing is going to be equal to 0 so this is different than what's going on over here so I definitely would not pick that let's do one more which of the following expressions are equivalent to a times negative 10 plus 11 well once again my you know the instant when you see something like as Y but I'd love to distribute that a it's just sitting there so let's do that a times negative 10 would be negative 10 times a or negative 10 a and then a times 11 so it's going to be plus a times 11 is the same thing as 11 times a which we could write as 11a now which of these choices are that negative 10a plus 11a so this is negative 10a plus 11a so this one looks right now what about this one well here they just swap the order that if you put the 11a first you could write it this way you could write if you write the 11a first we could write 11a and we have instead of saying at negative 10a we just say minus 10 a once again I just took all I did is I took this thing and I put it out front so these two things are actually equivalent so I would select that one as well