 So here's the important thing. Again, everything we do in basic arithmetic translates very naturally into algebra, and in fact, algebra is just the generalization of arithmetic. So all of our properties hold commutative, associative, and distributive properties, and importantly, the arithmetic of algebraic expressions is very similar to the arithmetic of the whole numbers. In fact, I would go so far as to say it is essentially identical. And there's two important definitions that we do want to remember. First of all, our definition of multiplication, a times x, well that is the sum of x over and over again, a times. The other definition it's important to remember is our definition of exponents, and that's just a notational issue by analogy with our definition of multiplication where a times x is the sum of a x's. This definition x to power n is the product of n x's. Alright, so let's consider some of our basic algebraic operations. So for example, let's take this 5x plus 8x, and that's equal to 13x, and let's go ahead and prove this as a statement. So you know the rules of algebra say if you're going to add two terms that have the same thing, you add the coefficients, and you get that and so on and so on and so forth. And it's a mechanical operation that you do, but it really comes back to this question of what do we mean by 5x, what do we mean by 8x, and what do we mean by 13x. And if you know what we mean by these, if you know this definition of, in this case, multiplication, this is obvious. So from our definition of multiplication, we're adding together 5x and 8x, so I have 5x's and I have 8x's, and so when I put those together I have 5 plus 8x's here, and that gives me a total of 13x's, and there's my sum. And if I want to I can actually summarize this as a rule of algebra when you add two things together with the same term, add the coefficients, and keep the same thing at the end, but really what we're actually doing is we're just doing this breaking apart into our multiplication. An important observation here is the rule is not a proof. You cannot use the rule as a proof. The actual proof 5x plus 8x equals 13x is the portion that's in green. We're going to write out 5x and 8x and add them, and then I get our 13x. And part of the reason this is important is the most common error in adding algebraic terms is combining them when you shouldn't. So again, going back to our rule of algebra, add the coefficients, 5, oh well we have an x and y there, we're going to call that an xy. Well why does that not work? Well it's pretty easy to see why if we go back to what these mean. 3x is 3x's, 2y is 2y's. So this sum here, 3x and 2y, I can write it out this way. Now what is 5xy? Well that's 5xy's, so 5xy is this, and if you look at this, and if you look at this, they are not identical. They are different, and because they are not identical, then you cannot write them as equal to each other. So here's an explanation of why 3x plus 2y is not the same as this. These two expressions are entirely different. We cannot write the equals, and what we want to write down is the section that's in green. Alright, well let's consider a different problem. How about let's do a subtraction, 12x minus 5x. And again there's a bunch of common errors that show up when students try to do this problem. Again, subtract the coefficients, 12 minus 5 is 7, and then subtract the x's. x minus x is nothing, and so the most common error when doing this problem, 12x minus 5x, is to give the answer 7, no x. Well, it's obvious that that's not correct if we try to prove it. So let's consider our definition of multiplication. Again, what we have here is we have 12x's, and we're going to remove 5x's. And so what do we have left over? Well, we have 7x's left over, and we can write that as 7x. So again, this is a problem that asks you to prove something, so we do have to include what is the section in green as our complete proof. Well, again, we can go further. What about algebraic multiplication? So again, at some point you learn how to multiply two algebraic expressions, but let's go ahead and prove that 8x times 3x is 24x squared. And, well, here the multiplication is going to be done using commutativity and associativity, and importantly our definition of exponents. So let's go ahead and write that down by commutativity and associativity. I can rearrange the factors any way that I want to. This is 8 times x, this is 3 times x, so I can rearrange this 8 times 3x times x. And commutativity and associativity allows me to do this, and I can multiply 8 times 3 is 24, and the definition of exponents x times x is by definition x multiplied by itself twice. So this expression here can be rewritten 24 times x squared. And again, the important thing to note here is that the actual problem is a proof. So in order to prove it, we need to indicate how we are arriving at each of our steps. And so here we have commutativity and associativity gives us this ability to rewrite the expression. So there's our proof. So here's a somewhat more involved proof. Prove that x plus 3 times x plus 5 is x squared plus 8x plus 15. Again, this is a proof problem, so no algorithm, no method of multiplying these two is acceptable as a proof of the result. You cannot use foil. You cannot use even the area model for the product. This is actually a proof, which means we have to go back to the properties of the algebraic expressions. So we might use the distributive property. We do have the distributive property in general. So this x plus 3 times x plus 5. Well, I can distribute this product. This is x plus 3 times x, x plus 3 times 5 by the distributive property. I can rearrange this by commutativity. And again, I have something that is set up for the distributive property one more time. This is x times x, x times 3, 5 times x, 5 times 3 by the distributive property. By the definition of exponents, this x times x is x squared by commutativity. 3 times x and x times 3 are the same. And now I can, by my definition of multiplication, I have 3x here. I have another 5x here. I have a total of 8x there. And that gives me my final result. And again, it's a proof, so I need to include everything that's shown in green.