Example 5
Completion requirements
| Example 5 |
Write \(3\sqrt[{5}]{2}\) as an entire radical.
Step 1: Move the coefficient into the radicand by applying an exponent equal to the index of the radical. Here, the index is \(5\), and \(3\) is rewritten as the fifth root of \(3\) to the power of five.
\(3\sqrt[5]{2} = \sqrt[5]{{3^5}}\cdot \sqrt[5]{2}\)
Step 2: Because the index values of the two radicals are equal, multiply the radicands together, combining them into one radical.
\(\sqrt[5]{{3^5}}\cdot \sqrt[5]{2} = \sqrt[5]{{3^5\cdot 2}}\)
Step 3: Simplify.
\(\begin{align}
\sqrt[5]{{3^5\cdot 2}} &= \sqrt[5]{{243 \cdot 2}} \\
&= \sqrt[5]{{486}} \\
\end{align}\)
Therefore, \(3\sqrt[{5}]{2} = \sqrt[{5}]{486}\).
\(3\sqrt[5]{2} = \sqrt[5]{{3^5}}\cdot \sqrt[5]{2}\)
Step 2: Because the index values of the two radicals are equal, multiply the radicands together, combining them into one radical.
\(\sqrt[5]{{3^5}}\cdot \sqrt[5]{2} = \sqrt[5]{{3^5\cdot 2}}\)
Step 3: Simplify.
\(\begin{align}
\sqrt[5]{{3^5\cdot 2}} &= \sqrt[5]{{243 \cdot 2}} \\
&= \sqrt[5]{{486}} \\
\end{align}\)
Therefore, \(3\sqrt[{5}]{2} = \sqrt[{5}]{486}\).