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Simplify the rational expressions. Identify the non-permissible values.

1. \(\frac{{12mn^3 }}{{150m^3 n}}\)
   
   
   Solution

   \[\frac{{12mn^3 }}{{150m^3n}} = \frac{{2n^2}}{{25m^2}}, \thinspace m \ne 0, \thinspace n \ne 0\]

   
   
   
   
2. \(\frac{{20z^2 + 45z}}{{30z^3 - 5z}}\)
   
   
   Solution

   \[\begin{align}
    \frac{{20z^2 + 45z}}{{30z^3 - 5z}} &= \frac{{{\color{red} \cancel {\color{#444} 5z}} \left( {4z + 9} \right)}}{{{\color{red} \cancel {\color{#444} 5z}} \left( {6z^2 - 1} \right)}} \\
     &= \frac{{4z + 9}}{{6z^2 - 1}}, \thinspace z \ne 0,\thinspace \pm \sqrt {\frac{1}{6}}  \\
    \end{align}\]

   
   Note: to determine the NPVs, refer to the factored form prior to simplifying.
   
   

   \[\begin{align}
    z &\ne 0 \\
     \\
    6z^2 - 1 &\ne 0 \\
    6z^2 &\ne 1 \\
    z^2  &\ne \frac{1}{6} \\
    z &\ne \pm \sqrt {\frac{1}{6}}  \\
    \end{align}\]

   
   
   
   
3. \(\frac{{10abc + 18a^2 bc}}{{3a^2 b + 8a^2 bc}}\)
   
   
   Solution

   \[\begin{align}
    \frac{{10abc + 18a^2 bc}}{{3a^2 b + 8a^2 bc}} &= \frac{{2abc\left( {5 + 9a} \right)}}{{a^2 b\left( {3 + 8c} \right)}} \\
     &= \frac{{2c\left( {5 + 9a} \right)}}{{a\left( {3 + 8c} \right)}},\thinspace a \ne 0,\thinspace b \ne 0,\thinspace c \ne - \frac{3}{8} \\
    \end{align}\]

   
   Note: To determine the NPVs, refer to the factored form prior to simplifying.
   
   \(\begin{align}
    a &\ne 0 \\
    b &\ne 0 \\
     \\
    3 + 8c &\ne 0 \\
    8c &\ne - 3 \\ 
    c &\ne -\frac{3}{8} \\
    \end{align}\)
   
   
   
4. \(\frac{{6x^2 + 5x + 1}}{{2x^2 + 5x + 2}}\)
   
   
   Solution

   \[\begin{align}
    \frac{{6x^2 + 5x + 1}}{{2x^2 + 5x + 2}} &= \frac{{\left( {3x + 1} \right){\color{red} \cancel {\color{#444} \left( {2x + 1} \right)}}}}{{{\color{red} \cancel {\color{#444} \left( {2x + 1} \right)}} \left( {x + 2} \right)}} \\
     &= \frac{{3x + 1}}{{x + 2}},\thinspace x \ne -\frac{1}{2},\thinspace - 2 \\
    \end{align}\]

   
   Note: To determine the NPVs, refer to the factored form prior to simplifying.
   
   \(\begin{align}
    2x + 1 &\ne 0 \\
    2x &\ne - 1 \\
    x &\ne - \frac{1}{2} \\
     \\
    x + 2 &\ne 0 \\
    x &\ne - 2 \\
    \end{align}\)
   
   
   

For further information about simplifying rational expressions, see pp. 312 to 316 of Pre-Calculus 11.