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# Multiplication of Rational Expressions

## Multiply and reduce fractions with variables in the denominator

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Practice Multiplication of Rational Expressions
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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?

### Guidance

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

#### Example A

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

Solution: 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.

#### Example B

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

Solution: These rational expressions are monomials with more than one variable. Here, we need to remember the laws of exponents from earlier concepts. 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.

#### Example C

Multiply 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*}

Solution: 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*}

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

2xy3z5xyz23x2yz34x3y2z26x3y4z420x4y3z43y10x

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

### Guided Practice

Multiply the following expressions.

1. 4x28x10x315x25xx2\begin{align*}\frac{4x^2-8x}{10x^3} \cdot \frac{15x^2-5x}{x-2}\end{align*}

2. 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*}

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

1. 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*}

2. 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*}

3. \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*}

### Explore More

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*}

1. \begin{align*}\frac{8x^2y^3}{5x^3y} \cdot \frac{15xy^8}{2x^3y^5}\end{align*}
2. \begin{align*}\frac{11x^3y^9}{2x^4} \cdot \frac{6x^7y^2}{33xy^3}\end{align*}
3. \begin{align*}\frac{18x^3y^6}{13x^8y^2} \cdot \frac{39x^{12}y^5}{9x^2y^9}\end{align*}
4. \begin{align*}\frac{3x+3}{y-3} \cdot \frac{y^2-y-6}{2x+2}\end{align*}
5. \begin{align*}\frac{6}{2x+3} \cdot \frac{4x^2+4x-3}{3x+3}\end{align*}
6. \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*}

### Vocabulary Language: English

Rational Expression

Rational Expression

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