 So as long as we're dealing with real numbers, the distributive property applies. a times the quantity b plus or minus c is a times b plus or minus a times c. So let's say I want to expand square root of 5 times quantity 2 plus square root of 7. So we're dealing with real numbers so the distributive property applies. I'll distribute the square root of 5 among the two factors. So that's going to be square root of 5 times 2 plus square root of 5 times square root of 7. Now as long as a and b are non-negative, the product of square roots is the square root of a product. So this square root 5 times square root 7 is going to be the same as the square root of 5 times 7, otherwise known as the square root of 35. Or we can try another one. So again, as long as we're dealing with real numbers, the distributive property applies. So every term inside the parentheses is going to be multiplied by square root of 2. The product of square roots is the square root of a product. So let's rearrange our factors and simplify. We'll rearrange our factors and simplify. How about something like this? We can expand this like we expand polynomial products. So let's draw our grid. One side of the rectangle will be 2 plus the square root of 5, and the other will be 3 minus the square root of 7. We'll multiply. And since all of our radicals are different, we can't really do much more than record the results. So remember when we say something squared, we really mean whatever it is times itself. So we can do that in exactly the same way. One side is going to be 5 minus the square root of 3, and the other side is going to be 5 minus the square root of 3. Finding our individual products gives us our final answer. Now in all of this there is one special factor which we define as follows. The conjugate of an expression involving square roots has the same terms, but the operation is changed. So an addition becomes a subtraction, or a subtraction becomes an addition. So let's find the conjugate of 8 plus square root of 7. Definitions are the whole of mathematics. All else is commentary. So let's pull in our definition of a conjugate. And so the conjugate will have the same terms, 8 square root of 7, but the operation changes. Since the original expression was an addition, the conjugate will be a subtraction. Square root of 7 minus square root of 11 says the conjugate will have the same terms, but the operation changes. So the conjugate will have a square root of 7, a square root of 11, but since the original expression was a subtraction, the conjugate will be an addition. Or a square root of 3 minus 2, so again the conjugate will have the same terms, but the operation changes. So the conjugate will have a square root of 3, 2, but since the original expression was a subtraction, the conjugate will be an addition. So you might wonder who cares? Why is the conjugate important? And the reason the conjugate is important has to do with a useful expansion. Remember that when we multiply a plus b times a minus b, we get a squared minus b squared. And so if we multiply a radical expression by its conjugate, we can eliminate some of the radicals. So for example, if we want to multiply by the conjugate, square root 3 minus 2. So we'll take our square root 3 minus 2 and multiply by the conjugate, which will have the same terms, square root 3 and 2, but instead of being a subtraction, the conjugate will be an addition. So when we multiply by the conjugate, we're going to get the square of the first term minus the square of the second term. And so that'll be 3 minus 4 or negative 1.