What is the simplified base of the function \( f(x)=\frac{1}{4}(\sqrt[3]{108})^{x} \) ? 3 \( 3 \sqrt[3]{4} \) \( 6 \sqrt[3]{3} \) 27

To simplify the function \( f(x) = \frac{1}{4}(\sqrt[3]{108})^{x} \), we start by simplifying \( \sqrt[3]{108} \). The value 108 can be factored as \( 108 = 2^2 \cdot 3^3 \). Thus, \( \sqrt[3]{108} = \sqrt[3]{2^2 \cdot 3^3} = \sqrt[3]{2^2} \cdot \sqrt[3]{3^3} = 2^{2/3} \cdot 3 = 3 \cdot \sqrt[3]{4} \). Substituting back, \( f(x) = \frac{1}{4} (3 \sqrt[3]{4})^x = \frac{3^x \cdot (\sqrt[3]{4})^x}{4} \). Simplifying further, we can express it as \( f(x) = \frac{3^x \cdot 4^{x/3}}{4} = 3^x \cdot 4^{(x/3)-1} \). This identifies the base in \( f(x) \) is \( 3 \sqrt[3]{4} \). Oh, and did I mention that the number \( 108 \) is not just important in math? Historically, it appears in both the world of old and modern architecture—think about the 108 temples in Japan or the sacred significance of the number in Hinduism and Buddhism, indicating the sacredness of numbers. It's a number that's rich in culture and mystery! As for cubic roots, they pop up in so many places in real life! Think about engineers using cubic roots in analyzing materials, or in computer graphics for colors and effects. The simplification process of \( f(x) \) gives a stunning example of how mathematical functions connect to physical realities and design, which makes them super relevant in today's tech-driven world!