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Equivalent expressions with negative numbers (multiplication and division)

In this video, we figure out whether or not some expressions are equivalent. To do this, we think about properties of multiplication and division.

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Video transcript

- [Voiceover] 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 negative numbers, or see these properties applied just to make sure that we understand what's going on. So these exercises, these are all from Kahn Academy. Oops. So this first one says, which of the following expressions are equivalent to negative two times the quantity five minus three? Now you could, of course, just figure out that five minus three is two and then multiply that times negative two and you would get negative four. And you could see which of these is equal to negative four. 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 two. I could multiply it times five and then I could multiply it by, I could either do it as it's going to be negative two times five plus negative two times negative three. Or you could view it as negative two times five minus negative two times positive three. Now let me write those two things down. So you could do this as negative two times five, negative two times five plus negative plus negative two times negative three. You could view it that way. Or you could view it as negative two, negative two times five minus minus and if I'm putting a minus here then I'm gonna view this as a positive three, that we're subtracting a positive three. So minus negative two times positive three. Notice I either wrote the negative here and wrote a positive here, or I wrote the negative here and made this a positive three. But these are going to be equivalent. Either way I've distributed this negative two. Notice I have a negative two, negative two. And what are these going to be equal to? Well, negative two times five is negative 10 and then, and then negative two times negative three is positive six, or over here negative two times three is negative six, but then we subtract it so it's just gonna get positive, you're gonna get positive six either way. This right over here is positive six and this over here, subtracting a negative six would give you positive six. So you get negative ten plus six. And that's this choice right over here. Which of course does evaluate to negative four, which this expression does evaluate to. This one up here evaluates to negative 16. And of course I won't select this because I found an answer. Let's do 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, oh what do these evaluate to. We need to do a little bit of 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, oops, let me do that in a different color, times S times T. Times T. And do any of the choices look like that? Well almost. Instead of saying negative S times S, this says S times negative S. And because multiplication, once again, I'm not a big fan of using the word because it sounds complicated, but 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 and maybe the easiest way is 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 ST. 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 was 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 or I'm taking S times S. So this is not, this is not the same thing. Which of the following expressions are equal 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. Alright, select all that apply. So let's just try to manipulate this a little bit. So once again, multiplication, it's associative. I could, so it's negative X times, 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 it's associative. 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 you're gonna 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 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 Xs, if I multiply the Xs 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, I 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 squared. This thing right over here, 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 evaluate to? Now this one actually evaluates to a number. Because regardless of what X you pick, X minus X, that's going to be zero. Zero times anything is going to be zero. So this thing is going to be equal to zero. 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, you know the instinct when you see something like this well 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 10A. And then A times 11, so it's gonna 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 swapped the order. If you put the 11A first, you could write it this way. If you write the 11A first, we could write 11A and then we have, instead of saying it negative 10A, we just say minus 10A. 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.