- Factor the number \(54\) as \(27 \times 2\), where \(27 = 3^3\) is a perfect cube.
- Use the property \(\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}\) to simplify \(\sqrt[3]{54x}\) to \(3\sqrt[3]{2x}\).
- Add the terms \(3\sqrt[3]{2x} + 3\sqrt[3]{2x}\) to get the final result \(6\sqrt[3]{2x}\).
- The correct choice is the first one, \(\boxed{6\sqrt[3]{2x}}\).
Explanation
- Understanding and Analyzing the Problem
We are tasked with simplifying the expression \(\sqrt[3]{54x} + 3\sqrt[3]{2x}\) under the condition that \(x
eq 0\). The expression involves cube roots of algebraic terms. To simplify these, we look for factors that are perfect cubes within the radicands. A perfect cube is an integer or expression raised to the third power (e.g., \(1, 8, 27, 64, \dots\)). Using the property \(\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}\), we can break down the radicals into simpler components.
- Simplifying the Radicals
First, let's look at the term \(\sqrt[3]{54x}\). We can factor \(54\) to identify perfect cubes. Since \(54 = 27 \times 2\) and \(27 = 3^3\), we can rewrite the radical as:
\[\sqrt[3]{54x} = \sqrt[3]{27 \cdot 2 \cdot x}\]
Using the product property of radicals, this becomes:
\[\sqrt[3]{27} \cdot \sqrt[3]{2x}\]
Since \(\sqrt[3]{27} = 3\), we have:
\[\sqrt[3]{54x} = 3\sqrt[3]{2x}\]
This simplification reveals that the first term of the original expression is \(3\sqrt[3]{2x}\).
- Combining Like Terms and Final Conclusion
Now, substitute the simplified form back into the original expression:
\[\sqrt[3]{54x} + 3\sqrt[3]{2x} = 3\sqrt[3]{2x} + 3\sqrt[3]{2x}\]
Since both terms are now multiples of the same radical, \(\sqrt[3]{2x}\), we can combine them like terms:
\[3\sqrt[3]{2x} + 3\sqrt[3]{2x} = (3 + 3)\sqrt[3]{2x} = 6\sqrt[3]{2x}\]
Comparing this result to the given options, we find it corresponds to the first option: \(6\sqrt[3]{2x}\).
Examples
Understanding how to simplify radical expressions is essential in fields like physics and engineering, where complex formulas often need to be reduced to a manageable form. For instance, when calculating the volume of objects with varying dimensions, you might encounter expressions involving cube roots. Simplifying these expressions early on makes it much easier to perform further algebraic manipulations, solve equations, or interpret physical measurements. Mastering radical simplification allows you to bridge the gap between abstract mathematical expressions and practical, real-world calculations.