 Given an expression involving a radical, we can add, subtract, multiply, or divide. And nothing important changes if we allow variables to be part of the expression. For example, suppose we want to expand the product. We'll use our area model, and we'll let one side of our rectangle be 5 plus square root of x, and the other side will be x minus square root of 3. So we'll compute the areas of our four individual rectangles, and our area, our product, will be the sum of all four areas. And the only simplification we can really make is going to come from this product of square roots. Remember that provided that our radicands are both positive, the product of a square root is the square root of the product. And we could try to rationalize the numerator. So remember that we can rationalize a radical expression by multiplying by a conjugate. And remember if you're dealing with a set of square roots, the conjugate is going to be the same terms, but you're going to change additions to subtractions and vice versa. So multiplying numerator and denominator by the conjugate. And remember factored form is best. Don't expand unless you need to. Well in this case we are trying to rationalize the numerator, and so if you're trying to rationalize a numerator or denominator, expand to show you've actually eliminated the square roots. Now since we don't care what happens to the denominator, we'll leave that in factored form. But we do want to expand the numerator, and we'll get. And remember the numerator should be considered to be one term. And this means we can remove a common factor from numerator and denominator, and so our expression simplifies. It's sometimes convenient to incorporate a variable inside a radical. Our ability to do this relies on two ideas. Provided that a and b are greater than or equal to zero, then the product of the square roots is the square root of the product. And also provided that a is greater than or equal to zero, the square root of a squared is equal to a. And remember equals means replaceable, and so these can work in both directions. For example if we want to rewrite this as the square root of a single expression, so five we can bring under a radical as square root of five squared. And because we have a product of square roots, it's also the square root of the product. Or we can take something like this. So the first thing is to get the things that are outside the radical into a square root. One should also be inside a square root. One of the other properties of working with square roots is the square root of a quotient is the quotient of the square roots. And now we have a product of square roots. So we can rewrite that as the square root of a product, expand and simplify.