 Real numbers have some useful properties. If a, b, and c are real numbers, then addition is associative. And that means if we have an addition like a plus b plus c, we can add the first two things together, a plus b, or we can add the second two things together, b plus c, and we get the same result. Multiplication is also associative. If we multiply three things, a times b times c, we can multiply the first two things, a times b first, or we can multiply the last two things, b times c first, and we get the same result either way. Addition is also commutative, a plus b is the same as b plus a, and multiplication is likewise commutative, a times b is the same as b times a. These properties are useful because they help us to perform some arithmetic operations, for example, 7 plus 6 plus 4. Since 7 plus 6 is in parentheses, we must evaluate it first according to the order of operations. So we find 7 plus 6, and to that result, we're supposed to add 4. But because of the associativity of addition, we can regroup the addends. In this case, rather than doing 7 plus 6 first, let's do 6 plus 4 first, and then 7 plus 10 is. Now, associativity by itself isn't very useful, and commutativity by itself isn't very useful. But, like so many things in life, they are stronger together. Put together, they provide a very useful result. An addition can be rearranged in any desired order, and likewise, a multiplication can be rearranged in any desired order. Now, it is important to remember that you can only rearrange the addends of an addition, and you can only rearrange the factors of a multiplication. Rearrangement is particularly important when we're dealing with fractions, so let's simplify 11-thirds times 5-8 times 3-11's. And by commutativity and associativity, we can rearrange those factors. And the thing you might notice here is that we have 11-thirds, and we have 3-11's, and if they were right next to each other, their product would be much easier to calculate. So let's do that. We'll switch the order inside the parentheses because we can. We'll regroup because we can. And now, 11-thirds times 3-11's is, and since I'm multiplying by 1, I'll just be left with 5-8's. So far, we've dealt with expressions that are all multiplications, or all additions. What if we have both an addition and a multiplication? Could we use commutativity and associativity to rearrange them? So is it true that 5 times 3 plus 2 is the same as, so I don't know, 5 times 2 plus 3? No. And the thing to remember here is that you can only rearrange terms if the operations are all the same, all multiplications or all additions. However, we can do something, which is called the distributive property, for real numbers A, B, and C. A times the quantity B plus C is the same as AB plus AC. So for example, let's say we can rewrite and evaluate 5 times 7 plus 2. So the distributive property says that this multiplication 5 times can be distributed amongst the addition 7 and 2. And so we can do 5 times 7 and 5 times 2, then add to get our final answer.