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Multiplication of Rational Expressions

Multiply and reduce fractions with variables in the denominator

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Multiplication of Rational Expressions

Suppose you were playing a game on your cell phone in which you were randomly given two rational expressions and were asked to identify the product of the two expressions. If one of the expressions were \begin{align*} \frac {x^2+3x+2}{x-9}\end{align*}x2+3x+2x9 and the other expression were \begin{align*} \frac {x^2-10x+9}{x^2-4}\end{align*}x210x+9x24, would you be able to multiply them together? Could the product be simplified? In this Concept, you'll learn about the multiplication of rational expressions such as these.

Guidance

Because a rational expression is really a fraction, two (or more) rational expressions can be combined through multiplication and/or division in the same manner as numerical fractions. A reminder of how to multiply fractions is below.

For any rational expressions \begin{align*}a \neq 0, b \neq 0, c \neq 0, d \neq 0\end{align*}a0,b0,c0,d0,

\begin{align*}\frac{a}{b} \cdot \frac{c}{d}= \frac{ac}{bd}\\ \frac{a}{b} \div \frac{c}{d} \rightarrow \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}\end{align*}abcd=acbdab÷cdabdc=adbc

Example A

Multiply the following: \begin{align*}\frac{a}{16b^8} \cdot \frac{4b^3}{5a^2}\end{align*}a16b84b35a2.

Solution:

\begin{align*}\frac{a}{16b^8} \cdot \frac{4b^3}{5a^2} \rightarrow \frac{4ab^3}{80a^2 b^8}\end{align*}a16b84b35a24ab380a2b8

Simplify exponents using methods learned in previous Concepts.

\begin{align*}\frac{4ab^3}{80a^2 b^8}=\frac{1}{20ab^5}\end{align*}4ab380a2b8=120ab5

Example B

Simplify \begin{align*}9c^2 \cdot \frac{4y^2}{21c^4}\end{align*}9c24y221c4.

Solution:

\begin{align*}9c^2 \cdot \frac{4y^2}{21c^4} \rightarrow \frac{9c^2}{1} \cdot \frac{4y^2}{21c^4}\\ \frac{9c^2}{1} \cdot \frac{4y^2}{21c^4}=\frac{36c^2 y^2}{21c^4}\\ \frac{36c^2 y^2}{21c^4}=\frac{12y^2}{7c^2}\end{align*}9c24y221c49c214y221c49c214y221c4=36c2y221c436c2y221c4=12y27c2

Multiplying Rational Expressions Involving Polynomials

When rational expressions become complex, it is usually easier to factor and reduce them before attempting to multiply the expressions.

Example C

Multiply \begin{align*}\frac{4x+12}{3x^2} \cdot \frac{x}{x^2-9}\end{align*}4x+123x2xx29.

Solution:

Factor all pieces of these rational expressions and reduce before multiplying.

\begin{align*}\frac{4x+12}{3x^2} \cdot & \frac{x}{x^2-9} \rightarrow \frac{4(x+3)}{3x^2} \cdot \frac{x}{(x+3)(x-3)}\\ & \frac{4\cancel{(x+3)}}{3x^{\cancel{2}}} \cdot \frac{\cancel{x}}{\cancel{(x+3)}(x-3)}\\ & \frac{4}{3x} \cdot \frac{1}{x-3} \rightarrow \frac{4}{3x^2-9x}\end{align*}4x+123x2xx294(x+3)3x2x(x+3)(x3)4(x+3)3x2x(x+3)(x3)43x1x343x29x

Guided Practice

Multiply \begin{align*}\frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18}\end{align*}12x2x6x21x2+7x+64x227x+18.

Solution: Factor all pieces, reduce, and then multiply.

\begin{align*}\frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18} & \rightarrow \frac{(3x+2)(4x-3)}{(x+1)(x-1)} \cdot \frac{(x+1)(x+6)}{(4x-3)(x-6)}\\ \frac{(3x+2)\cancel{(4x-3)}}{\cancel{(x+1)}(x-1)} \cdot \frac{\cancel{(x+1)}(x+6)}{\cancel{(4x-3)}(x-6)} & \rightarrow \frac{(3x+2)(x+6)}{(x-1)(x-6)}\\ \frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18} &= \frac{3x^2+20x+12}{x^2-7x+6}\end{align*}

Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Multiplying and Dividing Rational Expressions (9:19)

In 1–10, perform the indicated operation and reduce the answer to lowest terms

  1. \begin{align*}\frac{x^3}{2y^3} \cdot \frac{2y^2}{x}\end{align*}
  2. \begin{align*}\frac{2x}{y^2} \cdot \frac{4y}{5x}\end{align*}
  3. \begin{align*}2xy \cdot \frac{2y^2}{x^3}\end{align*}
  4. \begin{align*}\frac{4y^2-1}{y^2-9} \cdot \frac{y-3}{2y-1}\end{align*}
  5. \begin{align*}\frac{6ab}{a^2} \cdot \frac{a^3b}{3b^2}\end{align*}
  6. \begin{align*}\frac{33a^2}{-5} \cdot \frac{20}{11a^3}\end{align*}
  7. \begin{align*}\frac{2x^2+2x-24}{x^2+3x} \cdot \frac{x^2+x-6}{x+4}\end{align*}
  8. \begin{align*}\frac{x}{x-5} \cdot \frac{x^2-8x+15}{x^2-3x}\end{align*}
  9. \begin{align*}\frac{5x^2+16x+3}{36x^2-25} \cdot (6x^2+5x)\end{align*}
  10. \begin{align*}\frac{x^2+7x+10}{x^2-9} \cdot \frac{x^2-3x}{3x^2+4x-4}\end{align*}

Mixed Review

  1. The time it takes to reach a destination varies inversely as the speed in which you travel. It takes 3.6 hours to reach your destination traveling 65 miles per hour. How long would it take to reach your destination traveling 78 miles per hour?
  2. A local nursery makes two types of fall arrangements. One arrangement uses eight mums and five black-eyed susans. The other arrangement uses six mums and 9 black-eyed susans. The nursery can use no more than 144 mums and 135 black-eyed susans. The first arrangement sells for $49.99 and the second arrangement sells for 38.95. How many of each type should be sold to maximize revenue?
  3. Solve for \begin{align*}r\end{align*} and graph the solution on a number line: \begin{align*}-24 \ge |2r+3|\end{align*}.
  4. What is true of any line parallel to \begin{align*}5x+9y=-36\end{align*}?
  5. Solve for \begin{align*}d: 3+5d=-d-(3x-3)\end{align*}.
  6. Graph and determine the domain and range: \begin{align*}y-9=-x^2-5x\end{align*}.
  7. Rewrite in vertex form by completing the square. Identify the vertex: \begin{align*}y^2-16y+3=4\end{align*}.

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