Rationalize the denominator. Assume that all expressions under radicals represent positive numbers. \( \sqrt[3]{\frac{7}{2}} \)

To rationalize the denominator of \( \sqrt[3]{\frac{7}{2}} \), you can rewrite it as \( \frac{\sqrt[3]{7}}{\sqrt[3]{2}} \). To eliminate the cube root in the denominator, multiply both the numerator and the denominator by \( \sqrt[3]{4} \) (since \( \sqrt[3]{2^2} = \sqrt[3]{4} \)). This gives you: \[ \frac{\sqrt[3]{7} \cdot \sqrt[3]{4}}{\sqrt[3]{2} \cdot \sqrt[3]{4}} = \frac{\sqrt[3]{28}}{\sqrt[3]{8}} = \frac{\sqrt[3]{28}}{2}. \] So, the final rationalized form is \( \frac{\sqrt[3]{28}}{2} \). What's pretty cool about working with cube roots is how they relate to real-world applications, like calculating volumes! For instance, if you know the volume of a cube, you can easily derive the side length by taking the cube root. This concept appears regularly in architecture and design, where dimensions must relate neatly to practical uses. If you're looking to dive deeper into roots, check out topics like the 'Laws of Exponents' or 'Rational Exponents'! Understanding how exponents work can significantly ease your work with roots and radical expressions, allowing you to manipulate and simplify expressions like a pro! Plus, it's a foundational tool essential for tackling algebraic equations.