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Simplify the rational expressions, and identify any non-permissible values.

1. \(\frac{{4c}}{{ab}} \cdot \frac{{5a^2 }}{{2bc^2 }}\)
   
   
   Solution

   NPVs:
   
   \(a \ne 0\)
   \(b \ne 0\)
   \(c \ne 0\)
   
   \(\begin{align}
    \frac{{4c}}{{ab}} \cdot \frac{{5a^2 }}{{2bc^2 }} &= \frac{{20a^2 c}}{{2ab^2 c^2 }} \\
     &= \frac{{10a}}{{b^2 c}}, a \ne 0, b \ne 0, c \ne 0 \\
    \end{align}\)
   
2. \(\frac{{7\left( {q - 2} \right)}}{{\left( {p + 2} \right)}} \cdot \frac{{3\left( {p - 2} \right)}}{{21\left( {q - 2} \right)}}\)
   
   
   Solution

   NPVs:
   
   

   \(\begin{align}
    p + 2 &\ne 0 \\
    p &\ne -2 \\
    \end{align}\)

   \(\begin{align}
    q - 2 &\ne 0 \\
    q &\ne 2 \\
    \end{align}\)

   
   

   \[\begin{align}
    \frac{{{\color{red} \cancel {\color{#444}{7}}} {\color{red}\cancel {\color{#444}{\left( {q - 2} \right)}}}}}{{\left( {p + 2} \right)}} \cdot \frac{{3\left( {p - 2} \right)}}{{{\color{red} \cancel {\color{#444}{21}^3}} {\color{red}\cancel {\color{#444}{\left( {q - 2} \right)}}}}} &= \frac{1}{{p + 2}} \cdot \frac{{{\color{red}\cancel {\color{#444}{3}}} \left( {p - 2} \right)}}{\color{red} {\cancel {\color{#444}{3}}}} \\
     &= \frac{1}{{p + 2}} \cdot \frac{{p - 2}}{1} \\
     &= \frac{{p - 2}}{{p + 2}}, \thinspace p \ne - 2,\thinspace q \ne 2 \\
    \end{align}\]

   
   
   
3. \(\frac{{r^2 - 9r + 14}}{{r^2  - 16}} \cdot \frac{{2r + 8}}{{r^2 + 5r - 14}}\)
   
   
   Solution

   Factor all polynomials.
   
   

   \[\frac{{r^2 - 9r + 14}}{{r^2 - 16}} \cdot \frac{{2r + 8}}{{r^2 + 5r - 14}} = \frac{{\left( {r - 7} \right)\left( {r - 2} \right)}}{{\left( {r - 4} \right)\left( {r + 4} \right)}} \cdot \frac{{2\left( {r + 4} \right)}}{{\left( {r + 7} \right)\left( {r - 2} \right)}}\]

   
   NPVs:
   
   

   \(\begin{align}
    r - 4 &\ne 0 \\
    r &\ne 4 \\
    \end{align}\)
   

   \(\begin{align}
    r + 4 &\ne 0 \\
    r &\ne - 4 \\
    \end{align}\)
   

   \(\begin{align}
    r + 7 &\ne 0 \\
    r &\ne - 7 \\
    \end{align}\)
   

   \(\begin{align}
    r - 2 &\ne 0 \\
    r &\ne 2 \\
    \end{align}\)

   
   

   \[\begin{align}
    \frac{{r^2 - 9r + 14}}{{r^2 - 16}} \cdot \frac{{2r + 8}}{{r^2 + 5r - 14}} &= \frac{{\left( {r - 7} \right) {\color{red} \cancel {\color{#444}{\left( {r - 2} \right)}}}}}{{\left( {r - 4} \right) {\color{red}\cancel {\color{#444}{\left( {r + 4} \right)}}}}} \cdot \frac{{2{\color{red} \cancel {\color{#444}{\left( {r + 4} \right)}}}}}{{\left( {r + 7} \right) {\color{red} \cancel {\color{#444}{\left( {r - 2} \right)}}}}} \\
     &= \frac{{r - 7}}{{r - 4}} \cdot \frac{2}{{r + 7}} \\
     &= \frac{{2\left( {r - 7} \right)}}{{\left( {r - 4} \right)\left( {r + 7} \right)}}, \thinspace r \ne - 7, \thinspace \pm 4, \thinspace 2 \\
    \end{align}\]