We have rational functions whenever we have a fraction that has a polynomial in the numerator and/or in the denominator. An excluded value in the function is any value of the variable that would make the denominator equal to zero. To find the domain, list all the values of the variable that, when substituted, would result in a zero in the denominator.
When you guys start working with radical expressions, it's important that you know what a radical expression is. Radical expression just means it's an expression that has a square root sign. I call it a "square rootie" sometimes. I don't know why. It just comes out of my mouth. It's not an official term. So the other thing is that the thing that's under the radical sign, or under the "square rootie" is called a radicand. And you'll see that more when we start doing some problems. So when you're asked to simplify radical expressions, we have a really important property and here's what it is: If you have the square root of the product AB that's equal to the product of their individual square roots. It's equal to the square root of A times the square root of B. Just be really careful. This is only true as long as A and B are both positive and not 0. Let me show you an example. If I had the square root of 10, that's not something that you guys have probably worked with very much. The square root of 10 is equal to the square root of two times the square root of 5. If you don't trust me, grab yourself a calculator and check it out. The square root of 10 is the decimal of 3.16. The square root of two is the decimal 1.41. And I'm claiming that 3.16 is equal to 1.41 times whatever the square root of five is; 2.23. It's going to be a tiny bit off because I'm rounding, but you guys get the idea. This decimal times that decimal gives me that answer. And sometimes it doesn't make a whole lot of sense if you don't have a calculator handy, but it can be really useful in problems like this, like the square root of 18. I don't know what the square root of 18 is, but I do know that 18 is the equal to the product of nine times two. So the simplified form of the square root of 18 would be, let's see, square root of nine is three, times the square root of two. This would be my answer in simplified form. Simplified form means there are no perfect factors in the radicand, or no square numbers under the "square rootie", if that makes more sense to you. So when you're approaching these problems, it's really important that you're good at recognizing perfect square factors. I've gone through and listed all of the perfect squares for the numbers one through 15, like one times itself is one, two times two, three times three, four times four. Pretty much, you just have to memorize these. Get really comfortable with all the squares of the numbers one through 15, so that when you're doing these problems, you can recognize these factors. These are important numbers. One last thing, I want to leave you with before you start your homework problems is to watch out for this property. The negative square root of 144 is not the same thing as the square root of 144. That's really important. The negative square root of 144 would be 12. It's like I square rooted 144 and then negativize it, as opposed to this: The square root of 144, if you try it on your calculator, you'll see it says "Error". This is no real solution, no real number. There's no real number that when you multiply it by itself you get the answer 144. So watch out for that. Those are two really important distinctions.
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