Dividing Radicals and Rationalizing the Denominator - Problem 1

I’m asked to simplify this fraction. It’s not in the most reduced form because I have a radical in the denominator. You never want to have the square root in the bottom of the fraction. So what I’m going to do is go ahead and multiply top and bottom of a fraction by 1. That doesn’t change its value, but I’m going to be clever I’m not going to multiply by the number 1, I’m going to multiply by square root of 3x on top of square root of 3x which is equal to 1.

The reason why that’s clever is because now on the bottom, square root of 3x times square root of 3x is just regular 3x. I’m not going to have any square roots in the denominator anymore. I might have made the numerator more ugly, but the denominator is all cleaned up. That means my answer is going to be in simplified form.

Let’s look at the tops, 5 root 8 times root 3x is going to become 5 square roots of 24x. Before I decide that I’m done, I want to make sure that 24x cannot be reduced any further let me double check, Numbers that multiply into 24 looking for a square root, let’s see I could use 4 and 6. Square root of 4 times square root of 6 or I could write this as 2 root 6, so really this problem is going to be 5 times 2 root 6x. 2 root 6 came from the 24 part and there is the x. Simplify one step further by multiplying 5 and 2; this will be my final answer.

Be really careful because a lot of students are going to be tempted to cancel out those Xs, those Xs are not eligible to be reduced because one is under a square root sign and one is not. That means that those guys can’t be cancelled out.

This is my final answer. Don’t stop here because 24x under a radical or under a square root can still be reduced further to get to this.