Example 2
Completion requirements
| Example 2 |
Simplify the expression \(\left({\sqrt 2 + 3} \right)\left( {\sqrt 5 - 6} \right)\).
Step 1: Multiply the binomials following FOIL.
\(\begin{align}
\left( {\sqrt 2 + 3} \right)\left( {\sqrt 5 - 6} \right) &= \left( {\sqrt 2 \cdot \sqrt 5 } \right) + \left( {\sqrt 2 \cdot \left( { - 6} \right)} \right) + \left( {3 \cdot \sqrt 5 } \right) + \left( {3 \cdot \left( { - 6} \right)} \right) \\
&= \sqrt {10} + \left( { - 6} \right)\sqrt 2 + 3\sqrt 5 + \left( { - 18} \right) \\
&= \sqrt {10} - 6\sqrt 2 + 3\sqrt 5 - 18 \\
\end{align}\)
Step 2: Check if any radicals can be further reduced.
The radicals are all in simplest form.
\(\left( {\sqrt 2 + 3} \right)\left( {\sqrt 5 - 6} \right) = \sqrt {10} - 6\sqrt 2 + 3\sqrt 5 - 18\)
\(\begin{align}
\left( {\sqrt 2 + 3} \right)\left( {\sqrt 5 - 6} \right) &= \left( {\sqrt 2 \cdot \sqrt 5 } \right) + \left( {\sqrt 2 \cdot \left( { - 6} \right)} \right) + \left( {3 \cdot \sqrt 5 } \right) + \left( {3 \cdot \left( { - 6} \right)} \right) \\
&= \sqrt {10} + \left( { - 6} \right)\sqrt 2 + 3\sqrt 5 + \left( { - 18} \right) \\
&= \sqrt {10} - 6\sqrt 2 + 3\sqrt 5 - 18 \\
\end{align}\)
Step 2: Check if any radicals can be further reduced.
The radicals are all in simplest form.
\(\left( {\sqrt 2 + 3} \right)\left( {\sqrt 5 - 6} \right) = \sqrt {10} - 6\sqrt 2 + 3\sqrt 5 - 18\)