Given \( x>0 \) and \( y>0 \), select the expression that is equivalent to \[ \sqrt[3]{-8 x^{9} y^{11}} \] Answer \( 2 i x^{3} y^{\frac{11}{3}} \) \( 2 i x^{\frac{1}{3}} y^{\frac{3}{11}} \) \( -2 x^{\frac{1}{3}} y^{\frac{3}{11}} \) Submit Answer

To determine which expression is equivalent to \(\sqrt[3]{-8 x^{9} y^{11}}\), let's break it down step by step. ### Step 1: Simplify the Cube Root \[ \sqrt[3]{-8 x^{9} y^{11}} = \sqrt[3]{-8} \cdot \sqrt[3]{x^{9}} \cdot \sqrt[3]{y^{11}} \] ### Step 2: Evaluate Each Component 1. **Cube Root of \(-8\):** \[ \sqrt[3]{-8} = -2 \] 2. **Cube Root of \(x^9\):** \[ \sqrt[3]{x^{9}} = x^{9/3} = x^3 \] 3. **Cube Root of \(y^{11}\):** \[ \sqrt[3]{y^{11}} = y^{11/3} \] ### Step 3: Combine the Results \[ \sqrt[3]{-8 x^{9} y^{11}} = -2 \cdot x^3 \cdot y^{11/3} = -2 x^3 y^{11/3} \] ### Comparing with the Given Options Let's examine the provided options: 1. **\(2 i x^{3} y^{\frac{11}{3}}\)** This expression includes the imaginary unit \(i\), which is not necessary since the cube root of a negative number is real in this context. 2. **\(2 i x^{\frac{1}{3}} y^{\frac{3}{11}}\)** Similar to the first option, it unnecessarily includes \(i\) and has incorrect exponents. 3. **\(-2 x^{\frac{1}{3}} y^{\frac{3}{11}}\)** This expression has the correct sign but the exponents of \(x\) and \(y\) are inverted. ### Conclusion None of the provided options exactly match the simplified expression \(-2 x^3 y^{11/3}\). However, **Option 3** is the closest in terms of sign. It's possible there was a typographical error in the exponents. **Selected Answer:** \(-2 x^{\frac{1}{3}} y^{\frac{3}{11}}\)