Rationalize the denominator and simplify. \[ \sqrt[4]{\frac{2 b^{15}}{27 a}} \] Assume that all variables represent positive numbers. \( \square \)

To rationalize the denominator of \( \sqrt[4]{\frac{2 b^{15}}{27 a}} \), we first rewrite it: \[ \sqrt[4]{\frac{2 b^{15}}{27 a}} = \frac{\sqrt[4]{2 b^{15}}}{\sqrt[4]{27 a}} \] Next, we can separate the square roots: \[ \frac{\sqrt[4]{2} \cdot \sqrt[4]{b^{15}}}{\sqrt[4]{27} \cdot \sqrt[4]{a}} = \frac{\sqrt[4]{2} \cdot b^{\frac{15}{4}}}{\sqrt[4]{27} \cdot \sqrt[4]{a}} \] Now, let's simplify \( b^{\frac{15}{4}} = b^{3} \cdot b^{\frac{3}{4}} \). Finally, we need to rationalize the denominator \( \sqrt[4]{27} \), which can be expressed as \( 3^{\frac{3}{4}} \), and \( \sqrt[4]{a} = a^{\frac{1}{4}} \). Multiplying the numerator and the denominator by \( \sqrt[4]{a^3} \) will give us: \[ \frac{\sqrt[4]{2} \cdot b^{3} \cdot b^{\frac{3}{4}} \cdot \sqrt[4]{a^3}}{\sqrt[4]{27} \cdot \sqrt[4]{a} \cdot \sqrt[4]{a^3}} = \frac{\sqrt[4]{2} \cdot b^{3} \cdot b^{\frac{3}{4}} \cdot \sqrt[4]{a^3}}{\sqrt[4]{27 \cdot a^4}} = \frac{b^{3} b^{\frac{3}{4}} \sqrt[4]{2 a^3}}{3 \cdot a} \] Thus, the simplified expression with a rationalized denominator is: \[ \frac{b^{3} b^{\frac{3}{4}} \sqrt[4]{2 a^3}}{3 a} \] Or you can express it in its simplest form: \[ \frac{b^{3.75} \sqrt[4]{2 a^3}}{3 a} \] Feel free to substitute numerical values for \(a\) and \(b\) to test with real numbers!