- Factor the constant 648 into \(216 \times 3\), where 216 is \(6^3\).
- Factor the variable terms into powers of three: \(x^4 = x^3 \cdot x^1\) and \(y^8 = y^6 \cdot y^2 = (y^2)^3 \cdot y^2\).
- Extract the cube roots of \(6^3\), \(x^3\), and \((y^2)^3\), simplifying the expression outside the radical.
- The final simplified form is \(\boxed{18x^2y^2\sqrt[3]{3xy^2}}\).
Explanation
- Problem Analysis
We aim to simplify the expression \(3x\sqrt[3]{648x^4y^8}\). This involves identifying and extracting perfect cube factors from both the constant 648 and the variables \(x^4\) and \(y^8\). Recall that \(\sqrt[3]{a^3b} = a\sqrt[3]{b}\) for any real number \(a\), provided \(b\) is the remaining factor.
- Factoring and Rewriting the Expression
First, let's factorize the number 648. We know that \(6^3 = 216\). Dividing 648 by 216 gives \(648 / 216 = 3\). Thus, \(648 = 216 \times 3 = 6^3 \times 3\).
Next, examine the powers of the variables:
- For \(x^4\), the highest power that is a multiple of 3 is \(x^3\). Thus, \(x^4 = x^3 \times x^1\).
- For \(y^8\), the highest power that is a multiple of 3 is \(y^6\). Thus, \(y^8 = y^6 \times y^2 = (y^2)^3 \times y^2\).
Substituting these into the expression gives:
\[3x\sqrt[3]{6^3 \cdot 3 \cdot x^3 \cdot x \cdot (y^2)^3 \cdot y^2}\]
- Simplifying the Radical
Now, apply the cube root to each perfect cube factor individually:
\[3x \cdot \sqrt[3]{6^3} \cdot \sqrt[3]{x^3} \cdot \sqrt[3]{(y^2)^3} \cdot \sqrt[3]{3xy^2}\]
Evaluating the roots:
\[\sqrt[3]{6^3} = 6, \quad \sqrt[3]{x^3} = x, \quad \sqrt[3]{(y^2)^3} = y^2\]
Substituting these back, we get:
\[3x \cdot 6 \cdot x \cdot y^2 \sqrt[3]{3xy^2}\]
Multiplying the constant and variable terms outside the radical:
\[(3 \times 6) \times (x \times x) \times y^2 \sqrt[3]{3xy^2} = 18x^2y^2\sqrt[3]{3xy^2}\]
- Conclusion
Comparing our simplified result \(18x^2y^2\sqrt[3]{3xy^2}\) to the provided options, it matches the second option. Thus, the simplified form of the original expression is \(18x^2y^2\sqrt[3]{3xy^2}\). The final answer is \(\boxed{18x^2y^2\sqrt[3]{3xy^2}}\).
Examples
Simplifying radical expressions is essential in fields like physics and engineering, where complex formulas involve roots and powers. For example, when calculating the volume of a sphere given a radius expressed in radical form or dealing with resonant frequencies in electrical circuits, simplifying these expressions makes calculations manageable and allows for clearer interpretation of results. Mastering this technique helps in quickly reducing complex algebraic expressions to their most basic and useful forms.