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Rational expressions are multiplied together in much the same way that arithmetic fractions are multiplied together. To multiply rational numbers, we do the following:
Rational expressions are multiplied together using exactly the same three steps. Since rational expressions tend to be longer than arithmetic fractions, we can simplify the multiplication process by adding one more step.
Perform the following multiplications.
$$\frac{3}{4}\xb7\frac{1}{2}=\frac{3\xb71}{4\xb72}=\frac{3}{8}$$
$$\frac{8}{9}\xb7\frac{1}{6}=\frac{\stackrel{4}{\overline{)8}}}{9}\xb7\frac{1}{\underset{3}{\overline{)6}}}=\frac{4\xb71}{9\xb73}=\frac{4}{27}$$
$$\frac{3x}{5y}\xb7\frac{7}{12y}=\frac{\stackrel{1}{\overline{)3}}x}{5y}\xb7\frac{7}{\underset{4}{\overline{)12}}y}=\frac{x\xb77}{5y\xb74y}=\frac{7x}{20{y}^{2}}$$
$$\begin{array}{lll}\frac{x+4}{x-2}\xb7\frac{x+7}{x+4}\hfill & \hfill & \text{Divide out the common factor\hspace{0.17em}}x+4.\hfill \\ \frac{\overline{)x+4}}{x-2}\xb7\frac{x+7}{\overline{)x+4}}\hfill & \hfill & \text{Multiply numerators and denominators together}\text{.}\hfill \\ \frac{x+7}{x-2}\hfill & \hfill & \hfill \end{array}$$
$$\begin{array}{l}\begin{array}{lll}\frac{{x}^{2}+x-6}{{x}^{2}-4x+3}\xb7\frac{{x}^{2}-2x-3}{{x}^{2}+4x-12}.\hfill & \hfill & \text{Factor}\text{.}\hfill \\ \frac{\left(x+3\right)\left(x-2\right)}{\left(x-3\right)\left(x-1\right)}\xb7\frac{\left(x-3\right)\left(x+1\right)}{\left(x+6\right)\left(x-2\right)}\hfill & \hfill & \text{Divide out the common factors\hspace{0.17em}}x-2\text{\hspace{0.17em}and\hspace{0.17em}}x-3.\hfill \\ \frac{\left(x+3\right)\overline{)\left(x-2\right)}}{\overline{)\left(x-3\right)}\left(x-1\right)}\xb7\frac{\overline{)\left(x-3\right)}\left(x+1\right)}{\left(x+6\right)\overline{)\left(x-2\right)}}\hfill & \hfill & \text{Multiply}.\hfill \end{array}\\ \begin{array}{lllllllll}\frac{\left(x+3\right)\left(x+1\right)}{\left(x-1\right)\left(x+6\right)}\hfill & \hfill & \text{or}\hfill & \hfill & \frac{{x}^{2}+4x+3}{\left(x-1\right)\left(x+6\right)}\hfill & \hfill & \text{or}\hfill & \hfill & \frac{{x}^{2}+4x+3}{{x}^{2}+5x-6}\hfill \end{array}\\ \text{\hspace{0.17em}}\\ \text{Each of these three forms is an acceptable form of the same answer}.\end{array}$$
$$\begin{array}{l}\begin{array}{lll}\frac{2x+6}{8x-16}\xb7\frac{{x}^{2}-4}{{x}^{2}-x-12}.\hfill & \hfill & \text{Factor}\text{.}\hfill \\ \frac{2\left(x+3\right)}{8\left(x-2\right)}\xb7\frac{\left(x+2\right)\left(x-2\right)}{\left(x-4\right)\left(x+3\right)}\hfill & \hfill & \text{Divide out the common factors 2,\hspace{0.17em}}x+3\text{\hspace{0.17em}and\hspace{0.17em}}x-2.\hfill \\ \frac{\stackrel{1}{\overline{)2}}\overline{)\left(x+3\right)}}{\underset{4}{\overline{)8}}\overline{)\left(x-2\right)}}\xb7\frac{\left(x+2\right)\overline{)\left(x-2\right)}}{\overline{)\left(x+3\right)}\left(x-4\right)}\hfill & \hfill & \text{Multiply}.\hfill \end{array}\\ \begin{array}{ccc}\frac{x+2}{4\left(x-4\right)}& \text{or}& \frac{x+2}{4x-16}\end{array}\\ \text{\hspace{0.17em}}\\ \text{Both these forms are acceptable forms of the same answer}.\end{array}$$
$$\begin{array}{lll}3{x}^{2}\xb7\frac{x+7}{x-5}.\hfill & \hfill & \text{Rewrite\hspace{0.17em}}3{x}^{2}\text{\hspace{0.17em}as\hspace{0.17em}}\frac{3{x}^{2}}{1}.\hfill \\ \frac{3{x}^{2}}{1}\xb7\frac{x+7}{x-5}\hfill & \hfill & \text{Multiply}.\hfill \\ \frac{3{x}^{2}\left(x+7\right)}{x-5}\hfill & \hfill & \hfill \end{array}$$
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