- Use the property of radicals to write the expression as a quotient of two roots: \(\frac{\sqrt[4]{3}}{\sqrt[4]{2x}}\).
- Rationalize the denominator by multiplying both the numerator and denominator by \(\sqrt[4]{(2x)^3}\) to make the radicand in the denominator a perfect fourth power.
- Simplify the resulting expression to \(\frac{\sqrt[4]{24x^3}}{2x}\).
- The correct option is \(\boxed{\frac{\sqrt[4]{24x^3}}{2x}}\).
Explanation
- Separating the Radical Expression
We are tasked with simplifying the radical expression \(\sqrt[4]{\frac{3}{2x}}\) for \(x > 0\). The expression involves a fourth root of a fraction. To simplify this, we can use the property of radicals that allows us to separate the fourth root of the numerator and the denominator:
\[\sqrt[4]{\frac{3}{2x}} = \frac{\sqrt[4]{3}}{\sqrt[4]{2x}}.\]
To simplify the expression further, we rationalize the denominator by eliminating the radical term from the bottom of the fraction.
- Rationalizing the Denominator
The denominator is \(\sqrt[4]{2x}\). To remove the radical, we want the term inside the fourth root to be a perfect fourth power. Since we have \((2x)^1\), we need to multiply it by \((2x)^3\) to obtain \((2x)^4\), because \(\sqrt[4]{(2x)^4} = 2x\). Therefore, we multiply both the numerator and the denominator by \(\sqrt[4]{(2x)^3}\):
\[\frac{\sqrt[4]{3}}{\sqrt[4]{2x}} \cdot \frac{\sqrt[4]{(2x)^3}}{\sqrt[4]{(2x)^3}} = \frac{\sqrt[4]{3 \cdot (2x)^3}}{\sqrt[4]{(2x)^4}}.\]
- Simplifying and Comparing to Options
Now we simplify the numerator and denominator separately. For the denominator:
\[\sqrt[4]{(2x)^4} = 2x.\]
For the numerator:
\[\sqrt[4]{3 \cdot (2x)^3} = \sqrt[4]{3 \cdot 8x^3} = \sqrt[4]{24x^3}.\]
Combining these, we get:
\[\frac{\sqrt[4]{24x^3}}{2x}.\]
Comparing this result to the given options, it matches the second option.
Examples
In architecture and engineering, simplifying radical expressions is essential when working with design formulas that involve ratios of areas or volumes represented by roots. For instance, if you are scaling a 3D model, the relationship between volumes and linear dimensions often involves cube or fourth roots. Efficiently simplifying these expressions allows for quicker calculations and cleaner designs, ensuring that measurements are accurate and easy to work with in complex mathematical models.