- Simplify \(\frac{\sqrt[4]{6}}{\sqrt[3]{2}}\) by converting it to exponent form: \(6^{1/4} \cdot 2^{-1/3}\).
- Using prime factorization, express the term as \(2^{1/4} \cdot 3^{1/4} \cdot 2^{-1/3} = 2^{-1/12} \cdot 3^{3/12}\).
- Write this as a single root of a fraction: \(\sqrt[12]{\frac{27}{2}}\).
- Verify that \(\sqrt[12]{\frac{27}{2}} = \frac{\sqrt[12]{55296}}{2}\) since \(55296 / 2^{12} = 55296 / 4096 = 13.5\).
- The correct option is \(\boxed{\frac{\sqrt[12]{55296}}{2}}\).
Explanation
- Simplifying the Radical Expression
The goal is to simplify the expression \(\frac{\sqrt[4]{6}}{\sqrt[3]{2}}\). We begin by converting the radical expressions to fractional exponent form using the rule \(\sqrt[n]{x} = x^{1/n}\). The numerator becomes \(6^{1/4}\) and the denominator becomes \(2^{1/3}\). Substituting these into the original expression, we have:
\[\frac{6^{1/4}}{2^{1/3}}\]
Since \(6 = 2 \times 3\), we can rewrite the numerator as \((2 \times 3)^{1/4} = 2^{1/4} \times 3^{1/4}\). Now the expression is:
\[\frac{2^{1/4} \times 3^{1/4}}{2^{1/3}}\]
Using the laws of exponents, we can combine the terms with base 2:
\[2^{1/4 - 1/3} \times 3^{1/4} = 2^{(3/12 - 4/12)} \times 3^{1/4} = 2^{-1/12} \times 3^{1/4}\]
To express this with a common radical index of 12, we adjust the exponent of 3:
\[2^{-1/12} \times 3^{3/12} = \frac{3^{3/12}}{2^{1/12}} = \left(\frac{3^3}{2}\right)^{1/12} = \sqrt[12]{\frac{27}{2}}\]
Now, we compare this with the provided options. We test the third option: \(\frac{\sqrt[12]{55296}}{2}\). We can rewrite the denominator 2 as \(\sqrt[12]{2^{12}} = \sqrt[12]{4096}\). Thus,
\[\frac{\sqrt[12]{55296}}{\sqrt[12]{4096}} = \sqrt[12]{\frac{55296}{4096}} = \sqrt[12]{13.5}\]
Since \(13.5 = \frac{27}{2}\), the expression \(\sqrt[12]{\frac{27}{2}}\) is equivalent to \(\frac{\sqrt[12]{55296}}{2}\).
Examples
Imagine you are an engineer designing components for a high-performance engine where parts must have precise volumetric ratios. If two different types of volumetric measurements are taken, one being a fourth root and another being a cube root, you might need to combine them to find the overall ratio. Converting radicals to exponents allows engineers to manipulate these complex relationships easily, ensuring the final design specifications are accurate to the required level of precision.