Section 3.2
Multiplying and Dividing Fractions
When multiplying fractions, we multiply the numerators, multiply the denominators, and cancel out the common factors. This technique works well for simple problems, but is cumbersome for others. For example, if we multiply the numerators and multiply the denominators of the following expression, the result gets messy in a hurry. $$\frac{x^2-3x+2}{x^2-1}\cdot \frac{x^2+2x+1}{x^3-5x^2+6x}$$
Instead, a better technique is to factor first. Factor all the numerators and denominators and then cancel common factors.
Example:
a. \begin{align}
\frac{x^2-3x+2}{x^2-1}\cdot \frac{x^2+2x+1}{x^3-5x^2+6x}&=\frac{(x-2)(x-1)}{(x+1)(x-1)}\cdot \frac{(x+1)(x+1)}{x(x-3)(x-2)}\\
&=\frac{x+1}{x(x-3)}
\end{align}
To divide fractions, we will convert the division problem into a multiplication problem using the following property.
$$\frac{A}{B}\div\frac{C}{D}=\frac{A}{B}\cdot\frac{D}{C}$$
When dividing fractions, we multiply by the reciprocal of the second fraction and then simplify by factoring all the numerators and denominators and canceling out the common factors.
b. \begin{align}
\frac{x^2-3x+2}{x^3-4x}\div\frac{x^2+7x+12}{x^3+5x^2+6x}&=\frac{x^2-3x+2}{x^3-4x}\cdot \frac{x^3+5x^2+6x}{x^2+7x+12}\\
&=\frac{(x-2)(x-1)}{x(x+2)(x-2)}\cdot \frac{x(x+2)(x+3)}{(x+3)(x+4)}\\
&=\frac{x-1}{x+4}
\end{align}
It is appropriate to leave final results with the numerator and denominator in factored form, since then it is clear that no more reduction is possible.
Multiplying and dividing rational expressions 2: Multiplying and Dividing Rational Expressions 2
Multiplying and dividing rational expressions 3: Multiplying and Dividing Rational Expressions 3
Exercises
Simplify each of the following expressions.\begin{array}{l l}
1. \;\;\frac{{{a}^{2}}-7a-8}{2a+2}\cdot \frac{5}{a-8}\;\;& 2. \;\;\frac{8{{x}^{2}}}{{{x}^{2}}-25}\cdot \frac{3x+15}{4x}\;\; \\
3. \;\;\frac{10{{x}^{4}}}{24{{x}^{2}}}\cdot \frac{4{{y}^{2}}}{15x}\cdot \frac{12}{18y}\;\;& 4. \;\;\frac{4{{x}^{2}}+20x}{{{x}^{2}}+11x+30}\cdot \frac{{{x}^{2}}+2x-24}{2{{x}^{2}}-32}\;\; \\
5. \;\;\frac{{{x}^{2}}-5x+6}{9-{{x}^{2}}}\cdot \frac{2{{x}^{2}}+14x+24}{{{x}^{2}}-4x+4}\;\;& 6. \;\;\frac{10-40{{x}^{2}}}{4x-25}\cdot \frac{{{x}^{2}}-10x+25}{5-10x}\cdot {{\left( 2{{x}^{2}}-9x-5 \right)}^{-1}}\;\; \\
7. \;\;\frac{{{k}^{2}}-16k+64}{{{k}^{2}}-7k-8}\cdot \frac{8k+8}{64-{{k}^{2}}}\;\;& 8. \;\;\frac{{{x}^{2}}-5x+6}{{{x}^{2}}-9}\cdot \frac{2{{x}^{2}}+14x+24}{{{x}^{2}}-4x+4}\;\; \\
9. \;\;\frac{{{x}^{2}}-3x}{{{x}^{2}}+6x+9}\div \frac{3-x}{3+x}\;\;& 10. \;\;\frac{{{x}^{3}}-8}{{{x}^{2}}+4x+4}\div \frac{{{x}^{2}}+2x+4}{{{x}^{2}}-4}\;\; \\
11. \;\;\frac{2{{x}^{2}}+4x-30}{{{x}^{2}}+11x+30}\div \frac{4{{x}^{2}}-64}{2{{x}^{2}}+4x-48}\;\;& 12. \;\;\frac{{{x}^{3}}-4x}{2{{x}^{2}}+3x-2}\div \frac{3{{x}^{2}}-6x}{12{{x}^{2}}+6x-6}\;\; \\
13. \;\;\frac{4{{x}^{2}}-{{y}^{2}}}{2{{x}^{2}}+xy-{{y}^{2}}}\div \frac{2{{x}^{2}}-xy-{{y}^{2}}}{{{x}^{2}}-{{y}^{2}}}\;\;& 14. \;\;\frac{{{(x+6)}^{2}}}{x-1}\div \frac{36-{{x}^{2}}}{1-x}\;\; \\
15. \;\;\frac{{{a}^{2}}-4a+4}{{{a}^{2}}}\cdot \frac{a-1}{a}\div \frac{{{a}^{2}}-3a+2}{{{a}^{2}}}\;\;& 16. \;\;\frac{x+y}{{{x}^{2}}+{{y}^{2}}}\cdot \frac{x}{x-y}\div \frac{{{(x+y)}^{2}}}{{{x}^{4}}-{{y}^{4}}}\;\; \\
17. \;\;\frac{4{{x}^{2}}-1}{{{x}^{2}}-4x+3}\cdot \frac{6-2x}{2x-1}\div \frac{2x+1}{2{{x}^{2}}-3x+1}\;\;& 18. \;\;\frac{{{x}^{2}}-x-12}{{{x}^{2}}+2x-35}\div \left( \frac{{{x}^{2}}+3x-4}{{{x}^{2}}-25}\div \frac{{{x}^{2}}+6x-7}{{{x}^{2}}+8x+15} \right)\;\;
\end{array}