Example  6
Simplify the rational expression \(\frac{{4c}}{{15d}} \div \frac{{2c^2 }}{{21d}}\).  Identify any non-permissible values.


Step 1: Identify the NPVs.

\(d \ne 0\)

Step 2: Multiply by the reciprocal.

\[\frac{{4c}}{{15d}} \div \frac{{2c^2 }}{{21d}} = \frac{{4c}}{{15d}} \cdot \frac{{21d}}{{2c^2 }}\]

Step 3: Identify any new NPVs.

By rewriting as a product, \(c^2\) now appears in the denominator; therefore, \(c \ne 0\).

Step 4: Simplify.

\[\begin{align}
 \frac{{4c}}{{15d}} \div \frac{{2c^2 }}{{21d}} &= \frac{{4c}}{{15d}}\cdot \frac{{21d}}{{2c^2}} \\
  &= \frac{84cd}{30c^2d} \\
  &= \frac{{14}}{{5c}},\thinspace c \ne 0, \thinspace d \ne 0 \\
 \end{align}\]