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Algebra II: High School23 chapters | 203 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn what you can do to simplify your expression when you have a radical in the denominator. Learn what the single step solution to your problem is.

What do you think of when you hear the word irrational? I think of things that don't make sense. It's like a friend of mine telling me this story where he saw a giant blue cartoon character walk down the aisle of the grocery store. I know that doesn't make sense. Well, in math, an **irrational denominator** is similar because it has a radical in the denominator, which doesn't make sense in a fraction. In math, fractions should always have a number in the denominator. It shouldn't have radicals in them. Recall that radicals are those numbers inside the symbol that is also used by the square root. The square root is a radical with an index of 2. Because fractions shouldn't have radicals in the denominator, whenever we see a radical in the denominator we **rationalize the denominator**, or move the radical to the numerator. It is perfectly okay for a fraction to have a radical in the numerator.

How do we rationalize the denominator? If our denominator only has a radical, to move this radical to the numerator, we multiply the numerator and denominator by the same radical.

However, if our denominator has another number plus or minus a radical, then we need to multiply the numerator and denominator by the same denominator but with the opposite sign in between the two terms. We call this multiplying by the conjugate.

Let's see an example of each now.

In this first example, we have a denominator with a single radical.

Having this radical in the denominator makes our fraction not make sense, so we need to move it to the numerator. Because this is a single radical in the denominator, we can move it to the numerator by just multiplying both the numerator and denominator by the same radical. So, we multiply by the square root of 3 in the numerator and the denominator. We get 2 times the square root of 3 in the numerator, and we get a nice simple 3 in the denominator. Ah, now our fraction makes sense. We have rationalized our denominator. Because we can't simplify the square root of 3 further, we leave the numerator as 2 times the square root of 3.

Now, what if we have another number plus or minus our radical in the denominator?

What do we do now? This is where we multiply the numerator and denominator by the conjugate of the denominator. We take our denominator and rewrite it with the opposite sign in between the two terms. So we are multiplying by 5 plus the square root. Notice that instead of minus, we now have a plus.

Now, we go ahead and multiply out both the numerator and denominator. We use what we know about distributing our numbers, and we simplify.

When we multiply by our conjugate, we end up moving our radical. It does require a bit more math, but we end up with a fraction that makes sense. We have rationalized our denominator in this case, too.

What have we learned? We've learned that an **irrational denominator** is a fraction with a radical in the denominator. When we see this, we need to move the radical to the numerator because, in math speak, a radical in the denominator doesn't make sense. We call moving the radical to the numerator **rationalizing the denominator**. If we have just a single radical in the denominator, we move it to the numerator by just multiplying the numerator and denominator by the same radical. But, if we have another number plus or minus a radical, then we need to multiply the numerator and denominator by the conjugate, which is the denominator but with the sign in between the two terms opposite.

By the end of this lesson you should be able to rationalize the denominator of a fraction in a radical expression.

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Algebra II: High School23 chapters | 203 lessons

- What are the Different Types of Numbers? 6:56
- Graphing Rational Numbers on a Number Line 5:02
- Notation for Rational Numbers, Fractions & Decimals 6:16
- The Order of Real Numbers: Inequalities 4:36
- Finding the Absolute Value of a Real Number 3:11
- How to Rationalize the Denominator with a Radical Expression 3:52
- Understanding Numbers That Are Pythagorean Triples 3:41
- Go to Algebra II: Real Numbers

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