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Multiplication of Rational Expressions

Multiply and reduce fractions with variables in the denominator

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Multiplying Rational Expressions

The length of a rectangle is \begin{align*}\frac{2xy^3z}{5xyz^2}\end{align*}2xy3z5xyz2. The width of the rectangle is \begin{align*}\frac{3x^2yz^3}{4x^3y^2z^2}\end{align*}3x2yz34x3y2z2. What is the area of the rectangle?

Multiplying Rational Expressions

We take what you have learned previously a step further in this concept and multiply two rational expressions together. When multiplying rational expressions, it is just like multiplying fractions. However, it is much easier to factor the rational expressions before multiplying because factors could cancel out.

Let's multiply the following rational expressions.

  1. \begin{align*}\frac{x^2-4x}{x^3-9x} \cdot \frac{x^2+8x+15}{x^2-2x-8}\end{align*}x24xx39xx2+8x+15x22x8

Rather than multiply together each numerator and denominator to get very complicated polynomials, it is much easier to first factor and then cancel out any common factors.

\begin{align*}\frac{x^2-4x}{x^3-9x} \cdot \frac{x^2+8x+15}{x^2-2x-8} = \frac{x(x-4)}{x(x-3)(x+3)} \cdot \frac{(x+3)(x+5)}{(x+2)(x-4)}\end{align*}x24xx39xx2+8x+15x22x8=x(x4)x(x3)(x+3)(x+3)(x+5)(x+2)(x4)

At this point, we see there are common factors between the fractions.

\begin{align*}\frac{\cancel{x} \cancel{(x-4)}}{\cancel{x}(x-3) \cancel{(x+3)}} \cdot \frac{\cancel{(x+3)}(x+5)}{(x+2)\cancel{(x-4)}} = \frac{x+5}{(x-3)(x+2)}\end{align*}x(x4)x(x3)(x+3)(x+3)(x+5)(x+2)(x4)=x+5(x3)(x+2)

At this point, the answer is in factored form and simplified. You do not need to multiply out the base.

  1. \begin{align*}\frac{4x^2y^5z}{6xyz^6} \cdot \frac{15y^4}{35x^4}\end{align*}4x2y5z6xyz615y435x4

These rational expressions are monomials with more than one variable. Here, we need to remember the laws of exponents. Remember to add the exponents when multiplying and subtract the exponents when dividing. The easiest way to solve this type of problem is to multiply the two fractions together first and then subtract common exponents.

\begin{align*}\frac{4x^2y^5z}{6xyz^6} \cdot \frac{15y^4}{35x^4} = \frac{60x^2y^9z}{210x^5yz^6} = \frac{2y^8}{7x^3z^5}\end{align*}4x2y5z6xyz615y435x4=60x2y9z210x5yz6=2y87x3z5

You can reverse the order and cancel any common exponents first and then multiply, but sometimes that can get confusing.

  1. \begin{align*}\frac{4x^2+4x+1}{2x^2-9x-5} \cdot (3x-2) \cdot \frac{x^2-25}{6x^2-x-2}\end{align*}4x2+4x+12x29x5(3x2)x2256x2x2

Because the middle term is a linear expression, rewrite it over 1 to make it a fraction.

\begin{align*}\frac{4x^2+4x+1}{2x^2-9x-5} \cdot (3x-2) \cdot \frac{x^2-25}{6x^2-x-2} = \frac{\cancel{(2x+1)} \cancel{(2x+1)}}{\cancel{(2x+1)} \cancel{(x-5)}} \cdot \frac{\cancel{3x-2}}{1} \cdot \frac{\cancel{(x-5)}(x+5)}{\cancel{(3x-2)} \cancel{(2x+1)}} = x+5\end{align*}4x2+4x+12x29x5(3x2)x2256x2x2=(2x+1)(2x+1)(2x+1)(x5)3x21(x5)(x+5)(3x2)(2x+1)=x+5

Examples

Example 1

Earlier, you were asked to find the area of the rectangle. 

The area of the rectangle is length times width. So to find the area, multiply the two terms and simplify.

\begin{align*}\frac{2xy^3z}{5xyz^2} \cdot \frac{3x^2yz^3}{4x^3y^2z^2}\\ \frac{6x^3y^4z^4}{20x^4y^3z^4}\\ \frac{3y}{10x}\end{align*}2xy3z5xyz23x2yz34x3y2z26x3y4z420x4y3z43y10x

Therefore, the area of the rectangle is \begin{align*}\frac{3y}{10x}\end{align*}3y10x.

Multiply the following expressions.

Example 2

\begin{align*}\frac{4x^2-8x}{10x^3} \cdot \frac{15x^2-5x}{x-2}\end{align*}4x28x10x315x25xx2

\begin{align*}\frac{4x^2-8x}{10x^3} \cdot \frac{15x^2-5x}{x-2} = \frac{\cancel{2} \cdot 2 \cancel{x}\cancel{(x-2)}}{\cancel{2} \cdot \cancel{5x} \cdot \cancel{x} \cdot x} \cdot \frac{\cancel{5x}(3x-1)}{\cancel{x-2}} = \frac{2(3x-1)}{x}\end{align*}4x28x10x315x25xx2=22x(x2)25xxx5x(3x1)x2=2(3x1)x

Example 3

\begin{align*}\frac{x^2+6x-7}{x^2-36} \cdot \frac{x^2-2x-24}{2x^2+8x-42}\end{align*}x2+6x7x236x22x242x2+8x42

\begin{align*}\frac{x^2+6x-7}{x^2-36} \cdot \frac{x^2-2x-24}{2x^2+8x-42} = \frac{\cancel{(x+7)}(x-1)}{\cancel{(x-6)}(x+6)} \cdot \frac{\cancel{(x-6)}(x+4)}{2 \cancel{(x+7)}(x-3)} = \frac{(x-1)(x+4)}{2(x-3)(x+6)}\end{align*}x2+6x7x236x22x242x2+8x42=(x+7)(x1)(x6)(x+6)(x6)(x+4)2(x+7)(x3)=(x1)(x+4)2(x3)(x+6)

Example 4

\begin{align*}\frac{4x^2y^7}{32x^4y^3} \cdot \frac{16x^2}{8y^6}\end{align*}4x2y732x4y316x28y6

\begin{align*}\frac{4x^2y^7}{32x^4y^3} \cdot \frac{16x^2}{8y^6} = \frac{64x^4y^7}{256x^4y^9} = \frac{1}{4y^2}\end{align*}4x2y732x4y316x28y6=64x4y7256x4y9=14y2

Review

Determine if the following statements are true or false. If false, explain why.

  1. When multiplying two variables with the same base, you multiply the exponents.
  2. When dividing two variables with the same base, you subtract the exponents.
  3. When a power is raised to a power, you multiply the exponents.
  4. \begin{align*}(x+2)^2 = x^2 + 4\end{align*}(x+2)2=x2+4

Multiply the following expressions. Simplify your answers.

  1. \begin{align*}\frac{8x^2y^3}{5x^3y} \cdot \frac{15xy^8}{2x^3y^5}\end{align*}8x2y35x3y15xy82x3y5
  2. \begin{align*}\frac{11x^3y^9}{2x^4} \cdot \frac{6x^7y^2}{33xy^3}\end{align*}11x3y92x46x7y233xy3
  3. \begin{align*}\frac{18x^3y^6}{13x^8y^2} \cdot \frac{39x^{12}y^5}{9x^2y^9}\end{align*}18x3y613x8y239x12y59x2y9
  4. \begin{align*}\frac{3x+3}{y-3} \cdot \frac{y^2-y-6}{2x+2}\end{align*}3x+3y3y2y62x+2
  5. \begin{align*}\frac{6}{2x+3} \cdot \frac{4x^2+4x-3}{3x+3}\end{align*}62x+34x2+4x33x+3
  6. \begin{align*}\frac{6+x}{2x-1} \cdot \frac{x^2+5x-3}{x^2+5x-6}\end{align*}6+x2x1x2+5x3x2+5x6
  7. \begin{align*}\frac{3x-21}{x-3} \cdot \frac{-x^2+x+6}{x^2-5x-14}\end{align*}
  8. \begin{align*}\frac{6x^2+5x+1}{8x^2-2x-3} \cdot \frac{4x^2+28x-30}{6x^2-7x-3}\end{align*}
  9. \begin{align*}\frac{x^2+9x-36}{x^2-9} \cdot \frac{x^2+8x+15}{-x^2+11x+12}\end{align*}
  10. \begin{align*}\frac{2x^2+x-21}{x^2+2x-48} \cdot (4-x) \cdot \frac{2x^2-9x-18}{2x^2-x-28}\end{align*}
  11. \begin{align*}\frac{8x^2-10x-3}{4x^3+x^2-36x-9} \cdot \frac{5x+3}{x-1} \cdot \frac{x^3+3x^2-x-3}{5x^2+8x+3}\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 9.8. 

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Vocabulary

Rational Expression

A rational expression is a fraction with polynomials in the numerator and the denominator.

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