Example 2
Completion requirements
| Example 2 |
Multimedia |
Simplify \(\frac{{2 - \sqrt 5 }}{{3\sqrt {7p} + 2\sqrt 2 }}, p > 0\).
Step 1: Determine the conjugate of the denominator.
The conjugate of the denominator of this example is \(3\sqrt {7p} - 2\sqrt 2 \).
Step 2: Rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator, and then simplify.
Watch the coefficients in front of the square roots of the denominator. They must be squared as well!
The conjugate of the denominator of this example is \(3\sqrt {7p} - 2\sqrt 2 \).
Step 2: Rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator, and then simplify.
\(\begin{align}
\frac{{2 - \sqrt 5 }}{{3\sqrt {7p} + 2\sqrt 2 }} &= \frac{{\left( {2 - \sqrt 5 } \right)}}{{\left( {3\sqrt {7p} + 2\sqrt 2 } \right)}}\cdot \frac{{\left( {3\sqrt {7p} - 2\sqrt 2 } \right)}}{{\left( {3\sqrt {7p} - 2\sqrt 2 } \right)}} \\
&= \frac{{\left( {2 - \sqrt 5 } \right)\left( {3\sqrt {7p} - 2\sqrt 2 } \right)}}{{\left( {3\sqrt {7p} + 2\sqrt 2 } \right)\left( {3\sqrt {7p} - 2\sqrt 2 } \right)}} \\
&= \frac{{\left( {2 - \sqrt 5 } \right)\left( {3\sqrt {7p} - 2\sqrt 2 } \right)}}{{\left( {3\sqrt {7p} } \right)^2 - \left( {2\sqrt 2 } \right)^2 }} \\
&= \frac{{6\sqrt {7p} - 4\sqrt 2 - 3\sqrt {35p} + 2\sqrt {10} }}{{9\left( {7p} \right) - 4\left( 2 \right)}} \\
&= \frac{{6\sqrt {7p} - 4\sqrt 2 - 3\sqrt {35p} + 2\sqrt {10} }}{{63p - 8}} \\
\end{align}\)
\frac{{2 - \sqrt 5 }}{{3\sqrt {7p} + 2\sqrt 2 }} &= \frac{{\left( {2 - \sqrt 5 } \right)}}{{\left( {3\sqrt {7p} + 2\sqrt 2 } \right)}}\cdot \frac{{\left( {3\sqrt {7p} - 2\sqrt 2 } \right)}}{{\left( {3\sqrt {7p} - 2\sqrt 2 } \right)}} \\
&= \frac{{\left( {2 - \sqrt 5 } \right)\left( {3\sqrt {7p} - 2\sqrt 2 } \right)}}{{\left( {3\sqrt {7p} + 2\sqrt 2 } \right)\left( {3\sqrt {7p} - 2\sqrt 2 } \right)}} \\
&= \frac{{\left( {2 - \sqrt 5 } \right)\left( {3\sqrt {7p} - 2\sqrt 2 } \right)}}{{\left( {3\sqrt {7p} } \right)^2 - \left( {2\sqrt 2 } \right)^2 }} \\
&= \frac{{6\sqrt {7p} - 4\sqrt 2 - 3\sqrt {35p} + 2\sqrt {10} }}{{9\left( {7p} \right) - 4\left( 2 \right)}} \\
&= \frac{{6\sqrt {7p} - 4\sqrt 2 - 3\sqrt {35p} + 2\sqrt {10} }}{{63p - 8}} \\
\end{align}\)
Watch the coefficients in front of the square roots of the denominator. They must be squared as well!