Example 3
Completion requirements
| Example 3 |
Simplify the expression \(\left( {2\sqrt 7 + 3\sqrt 5 } \right)^2\).
Step 1: Multiply following FOIL.
\(\begin{align}
\left( {2\sqrt 7 + 3\sqrt 5 } \right)\left( {2\sqrt 7 + 3\sqrt 5 } \right) &= \left( {2\sqrt 7 } \right)^2 + \left( {2\sqrt 7\cdot 3\sqrt 5 } \right) + \left( {3\sqrt 5\cdot 2\sqrt 7 } \right) + \left( {3\sqrt 5 } \right)^2 \\
&= \left[ {2^2 \left( {\sqrt 7 } \right)^2 } \right] + \left[ {\left( {2\cdot 3} \right)\sqrt {7\cdot 5} } \right] + \left[ {\left( {3\cdot 2} \right)\sqrt {5\cdot 7} } \right] + \left[ {3^2 \left( {\sqrt 5 } \right)^2 } \right] \\
&= 4\cdot 7 + 6\sqrt {35} + 6\sqrt {35} + 9\cdot 5 \\
&= 28 + 12\sqrt {35} + 45 \\
&= 73 + 12\sqrt {35} \\
\end{align}\)
Step 2: Check if any radicals can be further reduced.
The radical is in simplest form.
\(\left( {2\sqrt 7 + 3\sqrt 5 } \right)^2 = 73 + 12\sqrt {35} \)
\(\begin{align}
\left( {2\sqrt 7 + 3\sqrt 5 } \right)\left( {2\sqrt 7 + 3\sqrt 5 } \right) &= \left( {2\sqrt 7 } \right)^2 + \left( {2\sqrt 7\cdot 3\sqrt 5 } \right) + \left( {3\sqrt 5\cdot 2\sqrt 7 } \right) + \left( {3\sqrt 5 } \right)^2 \\
&= \left[ {2^2 \left( {\sqrt 7 } \right)^2 } \right] + \left[ {\left( {2\cdot 3} \right)\sqrt {7\cdot 5} } \right] + \left[ {\left( {3\cdot 2} \right)\sqrt {5\cdot 7} } \right] + \left[ {3^2 \left( {\sqrt 5 } \right)^2 } \right] \\
&= 4\cdot 7 + 6\sqrt {35} + 6\sqrt {35} + 9\cdot 5 \\
&= 28 + 12\sqrt {35} + 45 \\
&= 73 + 12\sqrt {35} \\
\end{align}\)
Step 2: Check if any radicals can be further reduced.
The radical is in simplest form.
\(\left( {2\sqrt 7 + 3\sqrt 5 } \right)^2 = 73 + 12\sqrt {35} \)