 So, what can you do with a radical? How about adding and subtracting radicals? And this is part of a general mathematical process. The third question a mathematician asks about a new object is, what can we do with this? Once we have a radical expression, we need to consider the arithmetic of the expression. And some useful ideas to keep in mind. Arithmetic is bookkeeping. We're keeping track of how many of which items and algebra is generalized arithmetic. So let's take a deeper look at some arithmetic ideas. In arithmetic, we define multiplication first by the whole numbers. Let A and B be whole numbers. Then the product A times B is the sum of a whole bunch of B's, specifically A B's. So when I write 5 times 3, what I'm going to do is I'm going to add together 5 3's. And that gives me the product, so 5 times 3 is 15. In algebra, we generalize this definition. And the thing that we have to stick with is the number of times the factor B appears has to be a whole number. So let A be a whole number, then A times B is still going to be the sum of a whole bunch of B's, but this time we don't require that B be a whole number. So we can talk about 5 times X, well that's the sum of 5 X's. And remember, definitions are the whole of mathematics, all else is commentary. If you understand the definition of multiplication, you can add and subtract algebraic expressions. So let's pull back that definition of multiplication, and let's say I want to find 5 X plus 3 X. Well, 5 X is the sum of 5 X's, and 3 X is the sum of 3 X's. Now because the only operation over on the right-hand side is addition, we don't need the parentheses anymore. So let's count 1 X, 2 X's, 3 X's, and you get the general idea if we count the X's that we find we have 8 X's, and so 5 X plus 3 X is equal to 8 X. Well how about something like 5 square root of 3 plus 3 square root of 3? We can apply the same logic, 5 square root of 3 is the sum of 5 square root of 3, 3 square root of 3 is the sum of 3 square root of 3. Since we are only doing addition, we don't need the parentheses. And since this is a sum of square root of 3, we can just count the number that we have. 1, 2, 3, 4, 5, 6, 7, 8. There are 8 square root of 3, which is our sum. Now, once you know a little bit more mathematics, you can apply something a little bit more powerful than writing everything out, and so we could use the distributive property. So remember that for real numbers A, B, and C, A times the quantity B plus C is AB plus AC, and multiplication is commutative, so B plus C times A, well that's a BA plus CA. So 5 root 3 plus 3 root 3. Well here we have the principal square root of 3 as a common factor, and so we can remove that common factor, and rewrite, and we know how to find 5 plus 3, which gives us 8 square roots of 3. It's always a good idea in math and in life to stop every now and then and take stock of what we've already done and see if we can draw any parallels. So it's useful to draw parallels to other things you've done. So when we added fractions, 5 nths plus 3 nths, because they have the same denominator, we can add the numerators together, and their sum is going to be 5 plus 3 nths. Or if I add like terms, 5x plus 3x, because the variable portions are the same, the sum is 5 plus 3x. And we just found that 5 square root of 3 plus 3 square root of 3, because they have the same square roots, we can add the coefficients 5 plus 3 square roots of 3. What about subtraction? Well let's try to simplify 5 square root of 7 minus 2 square root of 7, and let's use two different methods. So first, let's write everything out. 5 square root of 7, well that's 1, 2, 3, 4, 5 square root of 7 minus 2 square root of 7, that means we're going to be subtracting 2 square root of 7. Now remember this minus means we're going to remove the square root of 7, so we're going to get rid of two of these square roots of 7. How about these two? And we'll be left with this, which is a bunch of square roots of 7, in fact it's 1, 2, 3 square roots of 7. Next, we'll use the distributive property. But wait, we can't quite do that yet, because the distributive property requires us to have terms added. And applying the distributive property to something that isn't an addition will void the warranty. Fortunately, we do know how to convert a subtraction into an addition. Any subtraction a minus b can be rewritten as a plus the additive inverse of b. So we'll rewrite this 5 square root of 7 minus 2 square root of 7. Well that's going to be 5 root 7 plus the additive inverse of 2 root 7. And now we have an addition so we can apply the distributive property. Both terms have a factor of square root 7, so we'll remove that common factor. Inside the parentheses we have 5 plus the additive inverse of 2. Well I know that if I add an additive inverse I get a subtraction, so that's the same as the subtraction 5 minus 2. And I can do that subtraction, that's going to give me 3 square roots of 7. What if I want to find square root of a plus square root of b where the radicands, the things underneath the square root symbol, are not the same? Well once again, it's useful to take stock of what we've done before. If I have variables and I try to add different variables, x plus y, I get x plus y, you can't combine unlike terms. And so it seems if I have a sum of 2 square roots, unless they're the same type of square root, I can't do anything with them. But wait, there are cases where we can add things that look a little bit different. So if I have fractions, 8 plus 1b, well I can add fractions if the denominators are different. And so the lesson to take away here is you can add fractions with different denominators if you change the denominators. And so the question is, maybe I can change these radicands to be the same, and that way I can add them. We'll take a look at that next.