Dividing Radicals (Advanced). In this video we’re going to look at how to divide radicals when you have the sum or difference of two terms in your denominator, or there’s a radical in your denominator. We do this using this rule: the difference of two squares. The difference of two squares says that a plus b, times, a minus b, is a squared minus b squared.

This is helpful because when you square a radical, the radical disappears. Let’s use it on this first example. We need to rationalize our denominator because right now it’s irrational. Any time you have the square root of a number that’s not a perfect square, then that is an irrational number.

We can’t have an irrational denominator, so to rationalize this denominator we use this difference of squares rule, by multiplying this denominator by its conjugate All the conjugate is, you take those two terms and, if you’re subtracting, then the conjugate would be those two terms added together, or the sum of those two terms. 4, plus, 2 square roots of 3, and 4, minus, 2 square roots of 3, are conjugates.

Now, we can’t just multiply the denominator by 4, plus, 2 square roots of 3, that would change the whole value of our expression, so we’re going to multiply the numerator also by 4, plus, 2 square roots of 3, because anytime you have an expression divided by itself, then that quotient is 1, so really what we’re multiplying by is 1, which means we’re not changing the value of this expression.

In your numerator we now need to distribute, so 4 times 4 is 16, plus, 4 times 2 squares of 3 is 8 square roots of 3. In your denominator we’re going to use this rule a plus b, times, a minus b. That’s what we have, a plus b, times, a minus b.

Again, these are conjugates of each other, and the rule says that we will get (our product will be) a squared minus B squared, so a squared, 4 squared would just be 16, and it’s always minus—it is called the difference of two squares—so minus, 2 square roots of 3, squared, so that’s 2, squared, and the square root of 3, squared. 2 squared is 4, the square root of 3, squared, is 3, (and keep in mind that even though it’s not there, there is a times sign, this is 2, times, the square root of 3) so when we square it, 2 squared 4, it would be 4 times 3, which is 12.

Now we can simplify that. We have 16, plus, 8 square roots of 3, divided by, 16 minus 12, which is 4, and we actually can continue simplifying this, and it might have been easier (or quicker) to instead of actually multiplying or distributing this 4 earlier, if we had just left it as 4 times the quantity (4, plus, 2 square roots of 3) because now all I’m going to do is actually just factor that 4 back out, so it’s 4 (my numerator) is 4, times, the quantity 4 plus 2 square roots of 3, divided by 4, because then what happens is 4 divided by 4 is 1, and our final answer is 4, plus, 2 square roots of 3.

Let’s look at one more problem where we use conjugates to simplify our expression. Again, our denominator is irrational because of those radicals, so we’re going to use this rule, difference of two squares, and multiply by the conjugate of our denominator. The conjugate of the square root of 5, plus, the square root of 2, is the square root of 5, minus, the square root of 2, so same two terms, the only difference is this plus and minus sign.

Then whatever we do to the denominator we do the exact same thing to the numerator, so that we have 1, were multiplying by 1. Then you can distribute here, or we can just leave it as negative 9, times the quantity, the square root of 5 minus the square root of 2, divided by—and then our denominator we’re going to use that rule, a squared minus b squared.

The square root of 5, squared, is 5, and it’s always minus the square root of 2 squared is 2. This is negative 9, times the quantity, the square root of 5 minus the square root of 2, divided by, 3, which we can then simplify. What is negative 9, divided by 3? That’s negative 3, times the quantity, the square root of 5 minus the square root of 2.

Then let’s go ahead and distribute that negative 3, so it’s negative 3 square roots of 5, plus, 3 square roots of 2 (can’t forget that 3) negative 3 times the square root of 5, negative 3 square roots of 5, negative 3 times the negative square root of 2 is a positive 3 square roots of 2. That’s how you divide radicals when you have two terms in your denominator.

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by Mometrix Test Preparation | Last Updated: June 13, 2019