- Convert the division problem into a fractional form: \(\frac{32 \sqrt[3]{18y}}{8 \sqrt[3]{3y}}\).
- Separate the expression into a product of constant coefficients and a quotient of radicals: \(\left(\frac{32}{8}\right) \cdot \left(\frac{\sqrt[3]{18y}}{\sqrt[3]{3y}}\right)\).
- Simplify the constant quotient to \(4\) and the radical quotient to \(\sqrt[3]{6}\) by canceling the common factor \(y\).
- Multiply the simplified terms to arrive at the final answer: \(\boxed{4 \sqrt[3]{6}}\).
Explanation
- Rewriting and Grouping the Expression
To simplify the expression \(32 \sqrt[3]{18y} \div 8 \sqrt[3]{3y}\) where \(y
eq 0\), we begin by expressing it as a fraction:
\[ \frac{32 \sqrt[3]{18y}}{8 \sqrt[3]{3y}} \]
This allows us to separate the constants from the radical components. By the rules of arithmetic, we can regroup the expression as:
\[ \left(\frac{32}{8}\right) \cdot \left(\frac{\sqrt[3]{18y}}{\sqrt[3]{3y}}\right) \]
We then simplify the constant portion, \(\frac{32}{8} = 4\). The expression now becomes:
\[ 4 \cdot \left(\frac{\sqrt[3]{18y}}{\sqrt[3]{3y}}\right) \]
- Simplifying Radicals
Next, we focus on simplifying the radical part \(\frac{\sqrt[3]{18y}}{\sqrt[3]{3y}}\). Using the property of radicals that \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}\), we can combine the terms under a single cube root:
\[ \sqrt[3]{\frac{18y}{3y}} \]
Given the condition \(y
eq 0\), we can cancel the \(y\) in the numerator and the denominator:
\[ \sqrt[3]{\frac{18}{3}} = \sqrt[3]{6} \]
Now, we multiply this result by the constant calculated in the previous step to obtain our final simplified form:
\[ 4 \cdot \sqrt[3]{6} = 4 \sqrt[3]{6} \]
- Final Conclusion
By simplifying the expression, we found it is equivalent to \(4 \sqrt[3]{6}\). Comparing this result with the given choices, we confirm that it matches the second option. Thus, the correct equivalent expression is \(4 \sqrt[3]{6}\).
Examples
Algebraic simplification of radical expressions is essential in fields like physics and engineering, where complex formulas involving roots are common. For example, when calculating the side length of a cube based on its volume, or determining resonance frequencies in electrical circuits, you often need to simplify expressions to make them manageable. By mastering these techniques, you can transform intimidating formulas into clear, solvable terms, ensuring accuracy and efficiency in your technical work.