Rationalize the Denominator 04:23 minutes

Video Transcript

Transcript Rationalize the Denominator

Meet Mellow Mike. He’s a happy fellow and quite content with his tranquil life living on the first floor of a two-story house except for one problem, his roommate, Radical Ruben. Mellow Mike wants Ruben to move out and go away because Ruben is way too radical for Mike’s taste. The chaos Ruben causes disrupts the harmony of their shared space. Dealing with Reuben is a lot like Rationalizing the Denominator in radical expressions with fractions. How, you ask? Let me show you.

Simplifying Radical Equations

When you simplify radical expressions in fractions, just like Mike, you want to get rid of the radical living on the first floor of the fraction or rather, in the denominator. Take a look at this fraction: two over root 3. How can we make root 3 go away? To solve this problem, let’s first review the Product Property of Square Roots. The property states the square root of ‘a’ times the square root of ‘b’ is equal to the square root of the product ‘ab.’

Let’s look at an example, the square root of 3 times the square root of 3 is the same as the square root of the product of 3 and 3 which is the same as the square root of 9 which is a perfect square - and equal to 3. How can we use this information to get rid of root 3 in the denominator of this fraction? Multiply the fraction by another fraction that is equal to one and comprised of the radical from the denominator - in this case, root 3 over root 3.

Lookey there, that made the radical in the denominator disappear! Of course, now we have a radical upstairs in the numerator, but we can live with that. But, if there’s a factor joined with the radical, what then? No problem, follow the same steps as before. Multiply the fraction by the new fraction equal to 1 that is comprised of the radical number from the denominator. Again, the radical moves from the downstairs to the upstairs, and that’s okay. Finally, if possible, simplify the fraction.

Sum in the denominator

What if the situation is more complicated? There’s a sum or difference with a radical in the denominator. Is this an impossible situation? No, just follow these steps. First, instead of multiplying the fraction by a new fraction equal to 1 comprised of the radical from the denominator, for this situation make the new fraction equal to 1 from the conjugate of the denominator. The conjugate?! What’s that you ask? In algebra, to write a conjugate, just multiply the second term of the second binomial by negative 1.

For example, for a + b, the conjugate is a – b. And just to help you understand why we do this, foil the two binomials the product is a-squared minus b-squared is equal to the difference of two squares because the two middle terms cancel each other out. Okay, so let's write a fraction equal to 1 that is comprised of the conjugate of 5 minus the square root of 7. Do the mathmand we're left with 15 plus 3 root 7 all divided by 18.

Looking at the constants and coefficients in each term, we find that they all have something in common: they're all divisible by 3. So we can factor out a 3 from the numerator and denominator and we're left with 5 plus the square root of 7 over 6. By multiplying the denominator by its conjugate, we're able to make the radical in the denominator disappear. It’s not magic, just math!

So just like in the example problems, Mellow Mike also got rid of his radical. I think you already know where he moved. Luckily for Radical Reuben, unlike Mellow Mike, Dancing Donna from the 2nd floor likes to raise the roof.