Example 5
Completion requirements
| Example 5 |
Simplify the expression \(\left( {\sqrt 3 + \sqrt r } \right)\left( {2\sqrt {5r} - \sqrt {2r} } \right)\). Indicate the restrictions of the variable.
Step 1: Identify the restrictions on \(r\).
Because the index is \(2\) (even), all radicands must be greater than or equal to zero.
\(r \ge 0, r \in \rm R\)
Step 2: Multiply the binomials following FOIL, and then check if any radicals can be further reduced.
\(\begin{align}
\left( {\sqrt 3 + \sqrt r } \right)\left( {2\sqrt {5r} - \sqrt {2r} } \right) &= 2\sqrt {15r} - \sqrt {6r} + 2\sqrt {5r^2 } - \sqrt {2r^2 } \\
&= 2\sqrt {15r} - \sqrt {6r} + 2r\sqrt 5 - r\sqrt 2 \\
\end{align}\)
Because the index is \(2\) (even), all radicands must be greater than or equal to zero.
\(r \ge 0, r \in \rm R\)
Step 2: Multiply the binomials following FOIL, and then check if any radicals can be further reduced.
\(\begin{align}
\left( {\sqrt 3 + \sqrt r } \right)\left( {2\sqrt {5r} - \sqrt {2r} } \right) &= 2\sqrt {15r} - \sqrt {6r} + 2\sqrt {5r^2 } - \sqrt {2r^2 } \\
&= 2\sqrt {15r} - \sqrt {6r} + 2r\sqrt 5 - r\sqrt 2 \\
\end{align}\)