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Simplifying Complex Rational Expressions

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  • 0:03 What Is a Complex…
  • 0:46 Rewriting the Problem
  • 1:36 Simplifying the Problem
  • 2:22 Another Example
  • 3:57 Lesson Summary
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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.

When it comes to fractions, things may seem to get really complicated and scary when you have a fraction on top of a fraction. But never fear, after watching this video lesson, you will be able to simplify these expressions like a pro.

What Is a Complex Rational Expression?

If you thought fractions were scary before, a complex rational expression will scare you even more! What is it? A complex rational expression is a fraction of fractions. So we have a fraction in our numerator and a fraction in our denominator. That is a complex rational expression. This is what we will be talking about in this video lesson, but don't run away just yet!

I will show you that it's not quite as scary as you think. I will show you just how easy it can be to simplify them and get them reduced to something that you can easily work with. The process that we will be using to tame these fraction monsters is a two-step process that involves rewriting our problem and then simplifying. Are you ready to get started? Let's go!

Rewriting the Problem

Let's start with a problem that includes only numbers, so we can see how the process works and how easy it is to use.

First problem.
Complex Rational

Don't get scared. This fraction monster won't bite. It might look tough and mean, but it's actually quite soft on the inside. Let's open this fraction up and see what's on the inside. We begin by rewriting our fraction. We know that fractions are actually division problems. So we actually have 4/5 divided by 10/6.

We also know that when we divide by a fraction, we can actually turn it into a multiplication problem by flipping the second fraction. So our 4/5 divided by 10/6 becomes 4/5 times 6/10. We have flipped our second fraction. So this is our rewritten problem:

Rewritten problem.
Complex Rational

Simplifying the Problem

Ah, multiplication! We can do multiplication easily with two fractions, can't we? Yes, we simply multiply across our numerator and multiply across our denominator. But before we do so, can we simplify any of these numbers? Is there a number in the numerator and the denominator that share a common factor?

Yes, there is. The 4 in the numerator and the 10 in the denominator both can be divided by 2. So we can simplify the 4 to 2 and the 10 to 5 by dividing both of these numbers by 2. So, now our problem is 2/5 times 6/5. Now we can perform our multiplication to get 12/25. This is our final answer, and we are done. That wasn't so bad, was it?

Another Example

Now, let's look at another example. This time we will see a problem with variables because many problems you will see will involve variables.

Second problem.
Complex Rational

Okay, so this problem is a little bit more complicated, since our denominator isn't a nice-looking fraction now. So, we need to turn our denominator into a fraction. We do this by adding our y/x and our 3. We recall that we need a common denominator before adding numerators. We see that our common denominator needs to be x, so the 3 needs to be turned into 3x/x.

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