Example 4
Completion requirements
| Example 4 |
Given the expression \(-4\sqrt[3]{{625r}} + \sqrt[3]{{40r^4 }}\), identify any restrictions on the variable, and then simplify the expression.
Step 1: Identify any restrictions on the variable.
Because the index is odd (three), there are no restrictions on \(r\).
\(r \in \rm R\)
Notice that although the index and the radicand are the same for each simplified radical, the coefficients of \(–20\) and \(2r\) cannot be further combined.
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.
\(\begin{align}
-4\sqrt[3]{{625r}} + \sqrt[3]{{40r^4 }} &= -4\sqrt[3]{{125\cdot 5r}} + \sqrt[3]{{8r^3\cdot 5r}} \\
&= -4\sqrt[3]{{5^3\cdot 5r}} + \sqrt[3]{{\left( {2r} \right)^3\cdot 5r}} \\
&= -4\cdot 5\sqrt[3]{{5r}} + 2r\sqrt[3]{{5r}} \\
&= -20\sqrt[3]{{5r}} + 2r\sqrt[3]{{5r}} \\
\end{align}\)
The expression cannot be simplified any further.
\(\begin{align}
-4\sqrt[3]{{625r}} + \sqrt[3]{{40r^4 }} &= -4\sqrt[3]{{125\cdot 5r}} + \sqrt[3]{{8r^3\cdot 5r}} \\
&= -4\sqrt[3]{{5^3\cdot 5r}} + \sqrt[3]{{\left( {2r} \right)^3\cdot 5r}} \\
&= -4\cdot 5\sqrt[3]{{5r}} + 2r\sqrt[3]{{5r}} \\
&= -20\sqrt[3]{{5r}} + 2r\sqrt[3]{{5r}} \\
\end{align}\)
The expression cannot be simplified any further.
Notice that although the index and the radicand are the same for each simplified radical, the coefficients of \(–20\) and \(2r\) cannot be further combined.