# Rationalize the Denominator 04:23 minutes

**Video Transcript**

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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.

**All Video Lessons & Practice Problems in Topic**Radical Expressions / Equations »