A **rational expression** is a ratio of two polynomial expressions.

Complete the table by finding the product of each row and column:

\dfrac{3}{2} | -\dfrac{1}{4} | \dfrac{1}{2} | -\dfrac{2}{5} | \dfrac{3}{8} | |
---|---|---|---|---|---|

\dfrac{3x}{2} | |||||

-\dfrac{1}{4x} | |||||

\dfrac{x}{2} | |||||

-\dfrac{2x}{5} | |||||

\dfrac{3}{8x} |

- Can each product be rewritten as a rational expression?
- What conclusions can we make about the products of rational expressions?
- Complete the table again, but this time find the quotient of each pair of rational expressions.
- What conclusions can we make about the quotients of rational expressions?

The product or quotient of two rational expressions always results in another rational expression, even if the denominator is simply 1.

To multiply two (or more) **rational expressions** together, we multiply the numerators to form the new numerator and multiply the denominators to form the new denominator - the same process used when multiplying fractions:\frac{A}{B} \cdot \frac{C}{D} = \frac{AC}{BD}

To divide two rational expressions, we multiply the first rational expression by the **reciprocal** of the second rational expression - the same process used when dividing fractions:\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} = \frac{AD}{BC}

Another way to represent division is with a complex fraction. The fraction in the numerator is divided by the fraction in the denominator.

\frac{A}{B} \div \frac{C}{D} = \dfrac{\frac{A}{B}}{\frac{C}{D}}

The same algebraic properties we use with real numbers apply to rational expressions:

Finding **common factors**, in particular the **greatest common factor (GCF)**, between any of the numerators and denominators can help us use the algebraic properties to simplify rational expressions.

We will need to assume that no denominator is equal to zero to avoid undefined expressions. We can do this by stating restrictions on the variables which will relate to restrictions on the domain of an associated rational function.

Fully simplify the expression, justifying each step. Write any restrictions on the variables.\frac{21a^3 b^6}{20 c^{2}} \cdot \frac{16 a^{2} c^{3}}{15 b^{6}}

Worked Solution

Fully simplify the expression, justifying each step. State any restrictions on the variables.\frac{10 x}{y^2 z} \div \frac{3 x^2 z}{10 y}

Worked Solution

Fully simplify the rational expression, justifying each step. Write any restrictions on the variables.\frac{x^2 + 3x - 10}{x^3 - 8} \cdot \frac{x^2 - 9}{x^2 + 2x - 15}

Worked Solution

Rewrite the complex fraction as a simplified rational expression, assuming no denominator equals zero:

\dfrac{\dfrac{3 - u}{14}} { \dfrac{u - 3}{3u^2}}

Worked Solution

Idea summary

By the definition of multiplying rational expressions, we know \dfrac{A}{B} \cdot \dfrac{C}{D}= \dfrac{AC}{BD}

We can use the algebraic properties of multiplication to find common factors and simplify the expressions.

Division of rational expressions can be rewritten as multiplication, where \dfrac{A}{B} \div \dfrac{C}{D} = \dfrac{A}{B} \cdot \dfrac{D}{C}

Rational expressions can also involve complex fractions which can be simplified using the same skills used with multiplication and division: \dfrac{\dfrac{A}{B}}{\dfrac{C}{D}}= \dfrac{A}{B} \div \dfrac{C}{D} = \dfrac{A}{B} \cdot \dfrac{D}{C} = \dfrac{AD}{BC}