Lesson 39: Using the Distributive Property to Simplify Rational Expressions

New Concept

A rational expression is an expression with a variable in the denominator.

What’s next

Next, you'll apply the distributive property to these expressions, combining a familiar rule with a new concept to expand your problem-solving toolkit.

Property

A rational expression is an expression with a variable in the denominator. A key rule is that the denominator cannot equal zero, so any variable value that causes this is not allowed.

Examples

• In the expression $\frac{5}{x}$, the restriction is $x \neq 0$.
• For $\frac{a+b}{c-3}$, the value $c=3$ is not allowed, so we state $c \neq 3$.
• In $\frac{m^2}{k(p+2)}$, we have two restrictions: $k \neq 0$ and $p \neq -2$.

Explanation

Think of rational expressions as fancy fractions that have variables in them! The most important rule in their world is that you can never, ever have a zero in the denominator. It's like trying to divide a pizza among zero friends—it just doesn't make sense! So, we always point out the values that are forbidden for the variables.

Property

To simplify expressions like $\frac{A}{B}\left(\frac{C}{D} + \frac{E}{F}\right)$, you distribute the outside term to each term inside the parentheses: $\frac{A \cdot C}{B \cdot D} + \frac{A \cdot E}{B \cdot F}$.

Examples

• $\frac{a^2}{b^3}\left(\frac{a^3}{b^2} + \frac{2b^4}{c}\right) = \left(\frac{a^2}{b^3} \cdot \frac{a^3}{b^2}\right) + \left(\frac{a^2}{b^3} \cdot \frac{2b^4}{c}\right) = \frac{a^5}{b^5} + \frac{2a^2b}{c}$
• $\frac{m}{n}\left(\frac{xy}{mk} - 3m^2\right) = \frac{m \cdot xy}{n \cdot mk} - \frac{m \cdot 3m^2}{n} = \frac{xy}{nk} - \frac{3m^3}{n}$

Explanation

This is like being a delivery person for math! The fraction outside the parentheses has to be delivered, or multiplied, to every single term waiting inside. After you've made all your deliveries, you just need to tidy up by using exponent rules to simplify each of the new fractions you've created. It's all about sharing the multiplication!

Property

When simplifying expressions, the final answer should not contain negative exponents. Use the rules $a^{-n} = \frac{1}{a^n}$ and $\frac{1}{a^{-n}} = a^n$ to fix them.

Examples

• $\frac{b^2}{d^{-4}}\left(\frac{3b^3}{d} - \frac{g^{-2}d}{b}\right) = \frac{3b^5}{d^{-3}} - \frac{b g^{-2}d}{d^{-4}} = 3b^5d^3 - \frac{bd^5}{g^2}$
• $\frac{n^{-2}}{m}\left(\frac{mx}{cn^{-4}} + 3n^{-1}p^{-3}\right) = \frac{n^{-2}mx}{mcn^{-4}} + \frac{3n^{-3}p^{-3}}{m} = \frac{n^2x}{c} + \frac{3}{mn^3p^3}$

Explanation

Think of a negative exponent as a sign that a term is on the wrong floor of a fraction! If a term with a negative exponent is in the numerator, move it to the denominator to make its exponent positive. If it's in the denominator, move it up to the numerator. They just need help getting to the right place!