 What about other types of radical expressions? It's useful to remember the following product. For any real numbers a and b, the product a plus b times a minus b is equal to a squared minus b squared. And this leads to a useful definition. We say that a plus b and a minus b are said to be conjugate expressions or conjugates. Remember definitions are the whole of mathematics, all else is commentary. One easy way to remember what a conjugate is is that the conjugates have the same terms, but in one they're added, and in the other they're subtracted. For example, let's find the conjugates of 40 plus 3, square root of 8 minus 7, and square root of 5 plus square root of 2. So definitions are the whole of mathematics, all else is commentary. Let's pull in our definition of conjugate. And again, the idea behind the conjugate is we'll have the same terms, but in one we'll add, and in the other we'll subtract. So the conjugate will have the same terms, but we change whatever the operation is. So the conjugate of 40 plus 3 has the same terms, 40 and 3. But since we're adding here, we'll want to subtract in the conjugate. So the conjugate will be 40 minus 3. The conjugate of square root of 8 minus 7 will have the same terms, square root of 8 and 7. But since we're subtracting here, our conjugate will be an addition. The conjugate of square root of 5 plus square root of 2 has the same terms, square root of 5 and square root of 2. But instead of an addition, we'll have a subtraction. So who cares about the conjugate? The reason that conjugation is so very important is you need to do it to speak Latin. Wait, wrong script. I mean, the reason that conjugation is so important in mathematics is that suppose either a or b or both involve a square root. Because the product a plus b times a minus b is a squared minus b squared, and when I multiply a square root by itself, I get the radicand, multiplying two conjugates will eliminate some of the square roots. For example, suppose we want to rationalize the numerator of the expression square root of 7 minus square root of 2 over 5. So remember that conjugate will have the same terms, square root of 7, square root of 2, but the operation will be different. Since we have a minus here, we want our conjugate to have a plus. And since we want to be able to use equals, we have to multiply numerator and denominator by the conjugate. Since we're trying to rationalize the numerator, let's go ahead and multiply out this mess. Square root 7 minus square root 2 times square root 7 plus square root of 2 is going to be square root 7 squared minus square root 2 squared, which we can simplify to 5. Into denominator, well, before we do anything, remember factored form is best. We had to multiply out the numerator because we didn't want to have any square roots in it, but in the denominator, we're already in factored form. Let's leave it in that form. So we'll leave our denominator as it is. The product of the factors in the numerator is equal to 5, so we'll write that. And now our numerator and denominator have a common factor of 5, so we can remove it. And that gives us our final answer, which has a rational numerator. One thing worth pointing out is that while we have rationalized the numerator, we haven't been able to suppress the radical expression. A version of it has appeared in the denominator. We could also try to rationalize the denominator, 8 over square root 3 minus 1. So the denominator, square root 3 minus 1, we can rationalize by multiplying by the conjugate. So the conjugate has the same terms, square root of 3 and 1, with a different operator. Instead of a minus, we'll write down a plus. Multiplying numerator and denominator by the conjugate, again factored form is best. So we'll leave the numerator as 8 times square root of 3 plus 1, but because we wanted to get rid of the radicals in the denominators, we do have to expand, which gives us a denominator of 2. We can simplify the fraction by removing common factors, and again the only thing that really matters is whether the numerator also has a factor of 2, and it does. We'll remove the common factor, giving us our final answer, and because factored form is best, we'll leave this in this form and not multiply it out.