## Multiplying and Dividing Rational Expressions

#### Aligned To Common Core Standard:

**High School** - HSA-APR.D.7

How to Multiply and Divide Rational Expressions - Rational numbers are always tricky to solve. Multiplication and division are two of the most basic operations in rational expressions. So, we try to learn that by using an example:
For example, let's simplify:
(8 – K) / (K^{2} - 64) ÷ (K – 8) / (K + 8)
So, whenever we have fractions divided by another fraction, we leave the first one alone and flip the second fraction as the division will now be converted into multiply.
(8 – K) / (K^{2} - 64) x (K + 8) / (K – 8)
If we pay close attention, we can see that the two factors have opposite signs in their numerators and denominators.
If we take "–" common from 8 – K, we get:
(K – 8) / (K^{2} - 64) x (K + 8) / (K – 8)
Using the rule of cross-multiplication we can cancel out K – 8. Also, we can expand K^{2} - 64. - (K + 8) / (K + 8) (K – 8) x (K + 8) / (K – 8)
Remainder: -1 / K – 8
Hence, this is how we divide and multiply the rational numbers.
This selection of worksheets and lessons teach students how to find products and quotients of a set of rational expressions.

### Printable Worksheets And Lessons

- Dividing Complex
Expressions Step-by-step Lesson- When I first saw this type
of problem as a kid, I remember think that I'm not in Kansas anymore.
This is the big time!

- Guided Lesson
- You will find multiplication and division spread across these
problems.

- Guided Lesson
Explanation - Step one is always to get everything in the same
form. With this skill it will require placing a one above or below
the current value.

- Practice Worksheet
- I tried to give you enough space to write on the paper with these
problems. Most kids feel that it is adequate room for them.

- Matching Worksheet
- Match the operations to the final value after performing all the
operations required to simplified.

#### Homework Sheets

I find this is one of those standards that kids either get on their first try or they need constant remediation on it.

- Homework 1 - To divide by a fraction, multiply by its reciprocal.
- Homework 2 - Rewrite the second expression as a fraction.
- Homework 3 - To divide, multiply by the reciprocal.

#### Practice Worksheets

See if you like how I fit the operations in there for these.

- Practice 1 - Multiply the numerators and multiply the denominators.
- Practice 2 - Divide and simplify your answer.
- Practice 3 - Watch out for all the extra operations here.

#### Math Skill Quizzes

I wrote individual directions for each problem to help students focus on their goal for each problem.

### Quick Tips for Solving Multiplication and Division of Rational Expressions Problems

A rational expression is a fractional math statement where either a numerator or denominator or both are polynomials. As we reviewed above, when you multiply these guys you treat them just like basic fractions which means that you find the product of the numerators and then you find the product of denominators. When finding the quotient of these expressions we use the same technique as multiplication, but before we do you turn the second fraction upside down (take the reciprocal). There are two trains of thought when performing these operations and it all revolves around when to simplify the fractions. I find that have mixed opinions on when you should reduce the fractions. Some teachers profess that you always start by reducing, other will understand this method, but have a slightly different take on it. Rather than reducing at the finish, they take a more case by case approach. If can see some operations that can be performed on the starting values, why fix what is not broken? The best summation of this method is to review the starting values and then go from there. If the values are hard to work with because they are large, then it is time to reduce before performing any operations. As you get more advanced, you may be confronted with large multi-term polynomials and quadratic expressions. In those cases you will need to factor the polynomials and locate common factors.