Rationalize the denominator. Assume that all variables represent positive real numbers. \( \frac{9}{\sqrt[3]{25 x^{2}}} \) \( \frac{9}{\sqrt[3]{25 x^{2}}}=\square \) (Type an exact answer, using radicals as needed.)

To rationalize the denominator of the expression \( \frac{9}{\sqrt[3]{25 x^{2}}} \), we need to eliminate the cube root in the denominator. We do this by multiplying the numerator and denominator by \( \sqrt[3]{25x} \), which will give us a perfect cube in the denominator. \[ \frac{9}{\sqrt[3]{25 x^{2}}} \cdot \frac{\sqrt[3]{25x}}{\sqrt[3]{25x}} = \frac{9\sqrt[3]{25x}}{\sqrt[3]{(25x^2)(25x)}} = \frac{9\sqrt[3]{25x}}{\sqrt[3]{625x^3}} = \frac{9\sqrt[3]{25x}}{5x} \] Thus, the rationalized form is \[ \frac{9\sqrt[3]{25x}}{5x}. \] Now, let's add some fun facts about this process! Did you know that rationalizing denominators dates back to ancient mathematicians? They understood the importance of simplifying expressions for easier calculations, paving the way for modern algebra. It’s not just about aesthetics; it’s about making those math problems friendlier and less intimidating! Additionally, rationalizing is used in various fields, including engineering and physics, where clarity and simplicity in calculations matter. It helps in solving equations or formulating designs, ensuring that professionals can communicate ideas and solutions without being bogged down by complex radicals! Understanding it not only helps in school but can be quite handy in tackling real-world problems too!