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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 2xy3z5xyz2\begin{align*}\frac{2xy^3z}{5xyz^2}\end{align*}. The width of the rectangle is 3x2yz34x3y2z2\begin{align*}\frac{3x^2yz^3}{4x^3y^2z^2}\end{align*}. 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. x24xx39xx2+8x+15x22x8\begin{align*}\frac{x^2-4x}{x^3-9x} \cdot \frac{x^2+8x+15}{x^2-2x-8}\end{align*}

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.

x24xx39xx2+8x+15x22x8=x(x4)x(x3)(x+3)(x+3)(x+5)(x+2)(x4)\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*}

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

x(x4)x(x3)(x+3)(x+3)(x+5)(x+2)(x4)=x+5(x3)(x+2)\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*}

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

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

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.

4x2y5z6xyz615y435x4=60x2y9z210x5yz6=2y87x3z5\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*}

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

1. 4x2+4x+12x29x5(3x2)x2256x2x2\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*}

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

4x2+4x+12x29x5(3x2)x2256x2x2=(2x+1)(2x+1)(2x+1)(x5)3x21(x5)(x+5)(3x2)(2x+1)=x+5\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*}

### 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.

2xy3z5xyz23x2yz34x3y2z26x3y4z420x4y3z43y10x\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*}

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

Multiply the following expressions.

#### Example 2

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

4x28x10x315x25xx2=22x(x2)25xxx5x(3x1)x2=2(3x1)x\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*}

#### Example 3

x2+6x7x236x22x242x2+8x42\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=(x+7)(x1)(x6)(x+6)(x6)(x+4)2(x+7)(x3)=(x1)(x+4)2(x3)(x+6)\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*}

#### Example 4

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

4x2y732x4y316x28y6=64x4y7256x4y9=14y2\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*}

### 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. (x+2)2=x2+4\begin{align*}(x+2)^2 = x^2 + 4\end{align*}

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

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

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### Vocabulary Language: English

Rational Expression

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

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