20. \( \begin{array}{l}\left(x^{2} y^{3}\right)^{\frac{3}{4}} \\ \left(x^{2}\right)^{\frac{3}{4}}\left(y^{3}\right)^{\frac{3}{4}} \\ x^{\frac{3}{2}} y^{\frac{9}{4}}\end{array} \)

Let's simplify the expression \(\left(x^{2} y^{3}\right)^{\frac{3}{4}}\) step by step. **Given Expression:** \[ \left(x^{2} y^{3}\right)^{\frac{3}{4}} \] **Step 1: Apply the Exponent to Each Factor Inside the Parentheses** Using the property of exponents \((ab)^n = a^n b^n\), we can distribute the exponent \(\frac{3}{4}\) to both \(x^2\) and \(y^3\): \[ \left(x^{2}\right)^{\frac{3}{4}} \left(y^{3}\right)^{\frac{3}{4}} \] **Step 2: Simplify Each Term by Multiplying the Exponents** Apply the power of a power rule \((a^m)^n = a^{m \cdot n}\): 1. For \(x^2\): \[ \left(x^{2}\right)^{\frac{3}{4}} = x^{2 \times \frac{3}{4}} = x^{\frac{6}{4}} = x^{\frac{3}{2}} \] 2. For \(y^3\): \[ \left(y^{3}\right)^{\frac{3}{4}} = y^{3 \times \frac{3}{4}} = y^{\frac{9}{4}} \] **Final Simplified Expression:** \[ x^{\frac{3}{2}} y^{\frac{9}{4}} \] **Summary of Steps:** 1. **Original Expression:** \(\left(x^{2} y^{3}\right)^{\frac{3}{4}}\) 2. **Distribute Exponent:** \(\left(x^{2}\right)^{\frac{3}{4}} \left(y^{3}\right)^{\frac{3}{4}}\) 3. **Simplify Exponents:** \(x^{\frac{3}{2}} y^{\frac{9}{4}}\) Therefore, the simplified form of \(\left(x^{2} y^{3}\right)^{\frac{3}{4}}\) is: \[ x^{\frac{3}{2}} y^{\frac{9}{4}} \]