Compare Your Answers
Completion requirements
Simplify the rational expressions, and identify any non-permissible values.
-
\(\frac{{\frac{{35pq}}{{r^2 }}}}{{\frac{{21p^2 }}{{qr}}}}\)
\[\frac{{\frac{{35pq}}{{r^2 }}}}{{\frac{{21p^2 }}{{qr}}}} = \frac{{35pq}}{{r^2 }} \div \frac{{21p^2 }}{{qr}}\]
NPVs:
\(r \ne 0\)
\(q \ne 0\)
\[\frac{{35pq}}{{r^2 }} \div \frac{{21p^2 }}{{qr}} = \frac{{35pq}}{{r^2 }} \cdot \frac{{qr}}{{21p^2 }}\]
New NPVs:
\(p \ne 0\)
\[\begin{align}
\frac{{35pq}}{{r^2 }} \div \frac{{21p^2 }}{{qr}} &= \frac{{35pq}}{{r^2 }} \cdot \frac{{qr}}{{21p^2 }} \\
&= \frac{{35pq^2 r}}{{21p^2 r^2 }} \\
&= \frac{{5q^2 }}{{3rp}}, \thinspace r \ne 0,\thinspace p \ne 0,\thinspace q \ne 0 \\
\end{align}\]
-
\(\frac{{3s + 7}}{{2\left( {s - 3} \right)}} \div \frac{5}{{s\left( {s - 3} \right)}}\)
NPVs:
\(\begin{align}
s - 3 &\ne 0 \\
s &\ne 3 \\
\end{align}\)\(s \ne 0 \)
\[\frac{{3s + 7}}{{2\left( {s - 3} \right)}} \div \frac{5}{{s\left( {s - 3} \right)}} = \frac{{3s + 7}}{{2\left( {s - 3} \right)}}\cdot \frac{{s\left( {s - 3} \right)}}{5}\]
There are no new NPVs.
\[\begin{align}
\frac{{3s + 7}}{{2\left( {s - 3} \right)}} \div \frac{5}{{s\left( {s - 3} \right)}} = \frac{{3s + 7}}{{2{\color{red}\cancel {\color{#444}{\left( {s - 3} \right)}}}}}\frac{{s{\color{red}\cancel {\color{#444}{\left( {s - 3} \right)}}}}}{5} \\
= \frac{{s\left( {3s + 7} \right)}}{{10}}, \thinspace s \ne 0, \thinspace 3 \\
\end{align}\]
-
\(\frac{{2t^2 + 11t + 5}}{{t^2 - 16}} \div \frac{{2t^2 + 3t + 1}}{{t^2 + t - 20}}\)
\[\frac{{2t^2 + 11t + 5}}{{t^2 - 16}} \div \frac{{2t^2 + 3t + 1}}{{t^2 + t - 20}} = \frac{{\left( {2t + 1} \right)\left( {t + 5} \right)}}{{\left( {t - 4} \right)\left( {t + 4} \right)}} \div \frac{{\left( {2t + 1} \right)\left( {t + 1} \right)}}{{\left( {t - 4} \right)\left( {t + 5} \right)}}\]
NPVs:
\(\begin{align}
t - 4 &\ne 0 \\
t &\ne 4 \\
\end{align}\)\(\begin{align}
t + 4 &\ne 0 \\
t &\ne - 4 \\
\end{align}\)\(\begin{align}
t + 5 &\ne 0 \\
t &\ne - 5 \\
\end{align}\)
\[\frac{{\left( {2t + 1} \right)\left( {t + 5} \right)}}{{\left( {t - 4} \right)\left( {t + 4} \right)}} \div \frac{{\left( {2t + 1} \right)\left( {t + 1} \right)}}{{\left( {t - 4} \right)\left( {t + 5} \right)}} = \frac{{\left( {2t + 1} \right)\left( {t + 5} \right)}}{{\left( {t - 4} \right)\left( {t + 4} \right)}} \cdot \frac{{\left( {t - 4} \right)\left( {t + 5} \right)}}{{\left( {2t + 1} \right)\left( {t + 1} \right)}}\]
New NPVs:
\(\begin{align}
2t + 1 &\ne 0 \\
t &\ne -\frac{1}{2} \\
\end{align}\)\(\begin{align}
t + 1 &\ne 0 \\
t &\ne - 1 \\
\\
\end{align}\)
\[\begin{align}
\frac{{\left( {2t + 1} \right)\left( {t + 5} \right)}}{{\left( {t - 4} \right)\left( {t + 4} \right)}} \div \frac{{\left( {2t + 1} \right)\left( {t + 1} \right)}}{{\left( {t - 4} \right)\left( {t + 5} \right)}} &= \frac{{{\color{red}\cancel {\color{#444}{\left( {2t + 1} \right)}}}\left( {t + 5} \right)}}{{{\color{red}\cancel {\color{#444}{\left( {t - 4} \right)}}}\left( {t + 4} \right)}}\cdot \frac{{{\color{red}\cancel {\color{#444}{\left( {t - 4} \right)}}}\left( {t + 5} \right)}}{{{\color{red}\cancel {\color{#444}{\left( {2t + 1} \right)}}}\left( {t + 1} \right)}} \\
&= \frac{{t + 5}}{{t + 4}} \cdot \frac{{t + 5}}{{t + 1}} \\
&= \frac{{(t + 5)^2}}{{(t + 4)(t + 1)}},\thinspace t \ne - 5, \pm 4, - 1, -\frac{1}{2} \\
\end{align}\]