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Completion requirements
Simplify the following expressions.
- \(\frac{2}{{3\sqrt {5t} - 4}}, t > 0\)
-
\(\frac{{3\sqrt 2 }}{{2\sqrt 5 + 4\sqrt 3 }}\)
The conjugate is \(2\sqrt 5 - 4\sqrt 3 \).
\(\begin{align}
\frac{{3\sqrt 2 }}{{\left( {2\sqrt 5 + 4\sqrt 3 } \right)}}\cdot \frac{{\left( {2\sqrt 5 - 4\sqrt 3 } \right)}}{{\left( {2\sqrt 5 - 4\sqrt 3 } \right)}} &= \frac{{6\sqrt {10} - 12\sqrt 6 }}{{\left( {2\sqrt 5 } \right)^2 - \left( {4\sqrt 3 } \right)^2 }} \\
&= \frac{{6\sqrt {10} - 12\sqrt 6 }}{{4\left( 5 \right) - 16\left( 3 \right)}} \\
&= \frac{{6\sqrt {10} - 12\sqrt 6 }}{{20 - 48}} \\
&= \frac{{6\sqrt {10} - 12\sqrt 6 }}{{ - 28}} \\
&= \frac{{6\sqrt 6 - 3\sqrt {10} }}{{14}} \\
\end{align}\)
The conjugate is \(3\sqrt {5t} + 4\).
\(\begin{align}
\frac{2}{{\left( {3\sqrt {5t} - 4} \right)}}\cdot \frac{{\left( {3\sqrt {5t} + 4} \right)}}{{\left( {3\sqrt {5t} + 4} \right)}} &= \frac{{6\sqrt {5t} + 8}}{{\left( {3\sqrt {5t} } \right)^2 - 4^2 }} \\
&= \frac{{6\sqrt {5t} + 8}}{{9\left( {5t} \right) - 16}} \\
&= \frac{{6\sqrt {5t} + 8}}{{45t - 16}},{\rm{ }}t > 0 \\
\end{align}\)
\(\begin{align}
\frac{2}{{\left( {3\sqrt {5t} - 4} \right)}}\cdot \frac{{\left( {3\sqrt {5t} + 4} \right)}}{{\left( {3\sqrt {5t} + 4} \right)}} &= \frac{{6\sqrt {5t} + 8}}{{\left( {3\sqrt {5t} } \right)^2 - 4^2 }} \\
&= \frac{{6\sqrt {5t} + 8}}{{9\left( {5t} \right) - 16}} \\
&= \frac{{6\sqrt {5t} + 8}}{{45t - 16}},{\rm{ }}t > 0 \\
\end{align}\)
| For further information about rationalizing the denominator, see pp. 287 – 288 of Pre-Calculus 11. |