Example 3
Completion requirements
| Example 3 |
Simplify the expression \(\sqrt {72} + 4\sqrt[3]{{3r}} - \sqrt[3]{{81r}} + \sqrt {162} \). Identify any restrictions on the variable.
Step 1: Identify any restrictions on the variable.
Because the index is odd (three), there are no restrictions on \(r\).
\(r \in \rm R\)
Step 2: Simplify individual radicals, where possible.
\(\begin{align}
\sqrt {72} + 4\sqrt[3]{{3r}} - \sqrt[3]{{81r}} + \sqrt {162} &= \sqrt {36\cdot 2} + 4\sqrt[3]{{3r}} - \sqrt[3]{{27\cdot 3r}} + \sqrt {81\cdot 2} \\
&= 6\sqrt 2 + 4\sqrt[3]{{3r}} - 3\sqrt[3]{{3r}} + 9\sqrt 2 \\
\end{align}\)
Step 3: Collect like terms to simplify the expression.
\(\begin{align}
{\color{blue}6\sqrt 2} {\color{red}+ 4\sqrt[3]{{3r}} - 3\sqrt[3]{{3r}}} {\color{blue}+ 9\sqrt 2} &= {\color{blue}\left( {6 + 9} \right)\sqrt 2} {\color{red}+ \left( {4 - 3} \right)\sqrt[3]{{3r}}} \\
&= {\color{blue}15\sqrt 2} +{\color{red} \sqrt[3]{{3r}}} \\
\end{align}
\)
Because the index is odd (three), there are no restrictions on \(r\).
\(r \in \rm R\)
Step 2: Simplify individual radicals, where possible.
\(\begin{align}
\sqrt {72} + 4\sqrt[3]{{3r}} - \sqrt[3]{{81r}} + \sqrt {162} &= \sqrt {36\cdot 2} + 4\sqrt[3]{{3r}} - \sqrt[3]{{27\cdot 3r}} + \sqrt {81\cdot 2} \\
&= 6\sqrt 2 + 4\sqrt[3]{{3r}} - 3\sqrt[3]{{3r}} + 9\sqrt 2 \\
\end{align}\)
Step 3: Collect like terms to simplify the expression.
\(\begin{align}
{\color{blue}6\sqrt 2} {\color{red}+ 4\sqrt[3]{{3r}} - 3\sqrt[3]{{3r}}} {\color{blue}+ 9\sqrt 2} &= {\color{blue}\left( {6 + 9} \right)\sqrt 2} {\color{red}+ \left( {4 - 3} \right)\sqrt[3]{{3r}}} \\
&= {\color{blue}15\sqrt 2} +{\color{red} \sqrt[3]{{3r}}} \\
\end{align}
\)