 In an expression like square root of x, we require the radicand to be non-negative. This means our rules for working with the square roots of real numbers also apply to working with the square roots of variable expressions. And one particularly important one is that for a and b greater than or equal to zero, the square root of a times the square root of b is equal to the square root of ab. Or the product of square roots is the square root of the product. And conversely, the square root of a product is the product of the square roots. Let's multiply and simplify this horrible mess. Now, we might begin by noticing that since we have a square root of 2x, we know that x itself must be non-negative. And so we can simplify. We can rearrange the factors. So let's put all of the things outside the square root together, the 3 and the 5, and let's put the square roots together. Square root of 2x, square root of 6x. And we know how to multiply 3 by 5, so let's go ahead and do that. So remember that as long as our radicands are non-negative, the product of square roots is the square root of the product. So square root 2x times square root 6x is the same as the square root of 2x times 6x, which simplifies, and let's see if we can simplify. So we'll try to factor 12x squared by isolating perfect square factors. So 12 is 4 times leftover 3, and x squared is a perfect square. The square root of a product is the product of the square roots. So we can rewrite this product. And since we started with a square root of x, we require that x be non-negative. And so we can say that square root of x squared is just equal to x. So square root of 4, that's 2. Square root of x squared is just x, and square root of 3 we're stuck with. And finally we'll do a last bit of simplification and multiply out the coefficients. How about something more complicated? Well again, since we have a square root of 12x cubed, the radicand 12x cubed must be non-negative. And remember, if n is odd, then x to the power n has the same sign as x, and x has the same sign as x to the n. Since we need 12x cubed to be non-negative, this means that x must be non-negative. And so we can simplify. We'll rearrange our factors. Let's multiply this 5 times 3 outside the radical. Now we can simplify before or after we multiply, but it might be easier to simplify first so that we're dealing with smaller things. So let's simplify our radicals first. Square root of 12x cubed will factor that by removing perfect square factors. Since x must be non-negative, we can say that square root of x squared is equal to x. And we can simplify our square roots. Square root of 2x to the seventh will simplify that by removing perfect square factors. Again, since x must be non-negative, we know that the square root of x squared is just x. So we can simplify all of the square roots of x squared. And since we have three factors of x, we'll just write that as x to the power 3. We'll rearrange our factors so all of the coefficients are multiplied together, and our square roots will be multiplied together. The product of square roots is the square root of the product. We'll multiply our coefficients together. And we can go one step further. This square root of 6x squared, well that's a perfect square, x squared times 6. And the square root of a product is the product of the square roots. Now since we know that x must be non-negative, so the square root of x squared is just x. And so we can simplify one step further. And finally we have a couple of factors outside the radical that we can multiply together to get our final simplified answer. And one more example. We can take this product of square roots, rearrange our factors. The product of square roots is the square root of a product. We can simplify the inside, find perfect square factors. And here we've assumed that all of our variables are non-negative. And because we've made that assumption, then the square root of x to power 10 can be reduced to x to power 5. And putting everything together gives us our final answer.