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

In this video lesson, you will learn how to divide and take the reciprocal of any rational expression. Learn the one easy step you take to be able to find your answer quickly and easily.

Rational Expressions

In this video lesson, we talk about rational numbers. What are they? They are numbers that can be written as the fraction of two integers. Remember that integers are whole numbers, both positive and negative. One way to think about rational numbers is when you want to share a certain number of things with a group of people, you have to divide what you have by the number of people. The math you get is a rational number.

For example, splitting 4 donuts among 3 people gives you 4/3, which is a rational number. Rational numbers include both fractions and your whole numbers because you can rewrite your whole numbers as a fraction being divided by 1. Your rational expressions, then, are math statements with rational numbers in them. So 4/3 by itself is both a rational number and a rational expression. (4/3) / (2/3) is a rational expression because it is a math statement with rational numbers in it.

Dividing a Rational

You might be looking at that last rational expression and wondering how in the world you would evaluate that kind of problem. What you are seeing is the division of two rational numbers. In this case, we see a fraction being divided by another fraction.

How do we evaluate this kind of problem? We evaluate it by turning our division problem into a multiplication by applying the one easy step of flipping the fraction we are dividing by. In our problem, the fraction we are dividing by is 2/3. We flip it by moving the denominator to the numerator and moving the numerator to the denominator. So 2/3 flipped becomes 3/2.

Now we can change our division into multiplication. So (4/3) / (2/3) turns into (4/3) * (3/2). Do we know how to multiply fractions? Yes, we simply multiply across. We multiply the numerators together, and we multiply our denominators together. So (4/3) * (3/2) becomes 12/6. Now we look at what we got and see if we can simplify it more. Yes, we can. 12/6 simplifies to 2. Our final answer is 2.

How can we remember this process? Well, if you think of a division problem as having an upper part and a lower part, then you can think of the lower part as being opposite the upper part. If the lower part is opposite, then to fix it, we just flip everything around. What is on top goes on the bottom, and what is on the bottom goes on the top. We only do the flipping on the lower part of our division problem. After we flip, there is no need for the division because we've made things right. We can now multiply.

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Taking the reciprocal is very closely related to division because it is 1 divided by our number. For example, the reciprocal of 4 is 1/4. This is easy to do when we have whole numbers. But what if we have a rational number that is a fraction? How do we take the reciprocal of one of these? For example, how do we take the reciprocal of (4/3)? We use the definition of reciprocal and we do 1 divided by our number. We get 1 / (4/3).

Now what? Well, we use what we know about dividing rational numbers and we flip the bottom rational number so that we can turn our problem into a multiplication problem. We get 1 * (3/4). Our answer, then, is 3/4. The reciprocal of 4/3 is 3/4. Do you notice something interesting here? Yes; the reciprocal of our rational number is simply the flipped version. To make it easy on yourself, just remember that the reciprocal of any rational number is simply the flipped version.

Examples

Let's look at some examples. The reciprocal of 5/4 is 4/5, the flipped version. Notice that the denominator is now in the numerator and the numerator is in the denominator.

What do you think the reciprocal of 6/7 is? That's right; it is 7/6.

What about division? How would you divide 9 by 1/3? 9 / (1/3). We flip the fraction in the denominator so we can turn it into a multiplication problem. We get 9 * (3/1). Our answer, then, is 27.

One more. (1/5) / (1/4). What do we do? Yes, we flip the bottom fraction and then turn the division into multiplication. (1/5) * (4/1). What does this equal? Multiplying straight across, we get 4/5. Our answer is 4/5.

Lesson Summary

Let's review what we've learned now. We learned that a rational number is a number that can be written as the fraction of two integers. A rational expression is a math statement with a rational number in it.

If we are dividing a rational number that happens to be a fraction, we flip the bottom fraction, the fraction we are dividing by so we can turn the problem into a multiplication problem. We then go ahead and multiply straight across. Since the reciprocal of a number is 1 divided by the number, the reciprocal of a rational number that is a fraction is simply the flipped version of it.

Learning Outcomes

By successfully finishing this lesson, you can expand your ability to:

Identify a rational expression

Solve a division problem that includes rational numbers

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