 When dealing with algebraic expressions, it's helpful to remember a couple of things. First, arithmetic is bookkeeping, how many of which objects? Second, algebra is generalized arithmetic. When we simplify an algebraic expression, we should still have the same number of each type of object. And finally, it's helpful to remember an important idea from the real numbers, associativity, and commutativity. For any real numbers A, B, and C, the product AB is equal to BA, and the product of AB with C is the same as the product of A with B and C. And in general, the factors of any product of real numbers can be rearranged in any order. So if we want to find 5 times 83 times 2, while the parentheses indicate that we should be multiplying 5 times 83 first, associativity and commutativity says we can rearrange this product by rearranging the factors. So we can use associativity to regroup, commutativity to switch the order, associativity to regroup, and that gives us a much easier product to start with. So for example, let's say we want to multiply 5x to the second by 3x to the fifth. By associativity and commutativity, it doesn't matter how we rearrange these factors. So this 5x squared 3x to the fifth, well, that's really 5 times x squared times 3 times x to the fifth. Associativity and commutativity mean we can rearrange these in any order that we want to. So let's rearrange the factors. We know how to multiply 5 and 3, so let's put those factors together. And arithmetic is bookkeeping, algebra is generalized arithmetic, we had an x squared and an x to the fifth, we still have them, which means we need to write those down as well. Now we know what 5 times 3 is, so let's compute that. x squared times x to the fifth, since those are exponents, we might consult our rules of exponents. And so here it's helpful to remember x to the second is the same as 2 factors of x multiplied together. It's x times x. Likewise, x to the fifth is the same as 5 factors of x multiplied together. It's x times x times x times x times x. Again, associativity means that we can group all of these factors together. And again, arithmetic is bookkeeping, algebra is generalized arithmetic, we want to keep track of how much we have, so we could write this long string of x's. But that's a bit tedious, and maybe we don't want to do that. However, definitions are the whole of mathematics, all else is commentary. We have 1, 2, 3, 4, 5, 6, 7 factors of x, so we can write this as x to the seventh. And again, arithmetic is bookkeeping, algebra is generalized arithmetic, we had an x to the second, x to the five to begin with, but it's the same as x to the seventh, so we should still have an x to the seventh, so we'll write that in. The process doesn't change significantly if we have more than one variable, so if we want to simplify 3xy times 5x squared by associativity and commutativity, because these are all multiplications, we can rearrange these in any order that we want to. So a convenient thing to do is to rearrange so that we're multiplying the numbers and then keep our variable factors together. We'll try to be organized and keep our x's together and our y's together. Part of the reason that that's useful is we know how to multiply two numbers, 3 times 5, and we can simplify one further step, x times x squared is x to the third. And again, arithmetic is bookkeeping, algebra is generalized arithmetic, we want to keep track of what we have, we had a y before, we should still have a y after. And that means we need to write it down.