Today we're going to talk about rationalizing denominators.
An expression with an irrational radical in its denominator is considered unsimplified. So for example, if you had the fraction 3 over the square root of 5, because the square root of 5 evaluates to be an irrational number, this fraction is considered unsimplified. So
I'm going to show you through some examples-- how you can simplify this expression so that we no longer have an irrational radical in the denominator of the fraction.
So let's talk about what a conjugate is. The conjugate of a binomial is just another binomial with the opposite sign between its two terms. So for example, if I have the square root of 3 plus 2-- this is a binomial. And so the conjugate of that is just going to be the square root of 3 minus 2. We just have the opposite sign between the two terms.
Similarly, the square root of 5 minus 1-- if I want to write the conjugate of that, that's going to be the square root of 5 plus 1.
And then if we have just a single term instead of a binomial-- the square root of 6-- we could think of the square root of 6 as square root of 6 plus 0, which would mean that its conjugate would be the square root of 6 minus 0. And the square root of 6 minus 0 is just the square root of 6. So the conjugate of just a single term radical is going to be that same radical.
So let's go ahead and do some examples to see why multiplying something by its conjugate will turn it from irrational to rational.
So let's take another look at why multiplying two conjugates together will give us a rational number.
So if I have the binomial square root of 2 plus 7, I know that my conjugate is going to be the square root of 2 minus 7. So I'm going to multiply those two kinds of conjugates together. And because they are two binomials, I'm going to use the FOIL method to multiply.
So starting with my first two terms, I've got the square root of 2 times the square root of 2, which I know the square roots cancel out, and that's just going to give me 2.
Then I've got my outside two terms-- the square root of 2 times 7. Since there's a minus sign in front of the 7, this is like negative numbers. So I'm going to write this as minus 7 square root of 2.
Then I've got my inside terms-- 7 and the square root of 2. So plus 7 square root of 2.
And then I'm going to multiply my last two terms. So positive 7 and this negative 7 will give me a minus or a negative 49.
So now when I simplify, I first see that because I have a minus 7 square root of 2 and a plus 7 square root of 2, those two things are opposites of each other, so they're just going to equal 0.
I'm going to bring down my other two numbers. And so now I just have 2 minus 0 minus 49. 2 minus 0 is just going to give us 2. And when I have 2 minus 49, that's going to give me a negative 47.
So you can see that when we start with a irrational number or a binomial and we multiply it by its conjugate, it eliminates the radicals or the irrational number, and we're left with just a rational number. So this is the idea behind rationalizing a denominator.
So let's go ahead and look at some examples where we have fractions and we use the conjugate to rationalize the denominator.
So for my first example, I just have a single term in my denominator, the square root of 6. I know that the conjugate of that is just the square root of 6. So when I multiply by the conjugate in the bottom of the fraction, I also want to make sure that I multiply it in the numerator. And that's because this is going to be equal to 1, will not actually change our original value.
So when I simplify the numerators and the denominators, 2 times the square root of 6 is going to give me 2 square root of 6, and the square root of 6 times the square root of 6 is just 6.
So this has no longer an irrational number in my denominator. And this example I could actually simplify one step farther because 2/6 is just 1/3. So I could write this as 1 square root of 6 over 3, or just the square root of 6 over 3.
Now let's do an example with a binomial. So it's the same thing. I'm going to multiply by the conjugate in the denominator of my fraction, so the square root of 5 minus 3 has a conjugate of square root of 5 plus 3. And I'll do the same thing in the top-- in the numerator of the fraction.
Now again, because I'm multiplying binomials, I'm going to use my FOIL method. So using the FOIL method, I'm going to start with my numerators. The square root of 8 times the square root of 5 is my first terms.
My outside times are 3 and the square root of 8. My inside terms are 2 and the square root of 5. And my last terms are 2 times 3.
Now for my denominator, I've got my first terms as the square root of 5 times the square root of 5. My outside terms are 3 and the square root of 5. My inside terms are a minus 3 and the square root of 5. And my last two terms to be multiplied are minus 3 and positive 3.
Now I'm going to start by simplifying my numerator. I know that using the product property, square root of 8 times the square root of five is just the square root of 40. I've got three square roots of 8 and two square roots of 5. Those cannot be combined together because they are different radicals. And then I can simplify 2 times 3 as 6. That's as simple as my numerator is going to get.
So my denominator, I know that the square root of 5 times the square root of 5 is just going to give me 5. Positive 3 square root 5 and a negative 3 square root of 5 will cancel out. And then I just have minus my 3 times 3, which is just 9.
So now I can just simplify my denominator one step farther. 5 minus 9 is going to give me a negative 4.
So when you have on a negative number in the denominator of your fraction, sometimes it's helpful just to put it in the beginning of the fraction to make it a little bit more clear.
So let's go over our key points from today. Make sure you get them in your notes if you don't already so you can refer to them later.
So we talked about the fact that rationalizing a denominator involves multiplying by a conjugate in both the denominator and the numerator of a fraction and then simplifying. So the reason that we do that is because having an irrational radical in the denominator of a fraction is not considered simplified.
And then, we looked at what a conjugate is. And so we saw that the conjugate of the square root of a plus b is just the square root of a minus b. We just use the opposite sign in between the two terms. So the conjugate of square root of a minus b is square root of a plus b.
And we saw that the conjugate of just a radical by itself is that same radical. So I hope that these key points in the examples that we did helped you understand a little bit more about rationalizing denominators.
Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.
The conjugate of a binomial is a binomial with the opposite sign between its terms.