((32x^frac (5)/(3)y^(2)/(3))^(1)/(5))((16x^-3y^2)^(1)/(2))

## Step 1: Simplify the Numerator<br />### Apply the power of a power rule \((a^m)^n = a^{m \cdot n}\) to simplify the numerator. <br />\[<br />(32 x^{\frac{5}{3}} y^{\frac{2}{3}})^{\frac{1}{5}} = 32^{\frac{1}{5}} \cdot x^{\frac{5}{3} \cdot \frac{1}{5}} \cdot y^{\frac{2}{3} \cdot \frac{1}{5}}<br />\]<br />Calculate each component:<br />- \(32^{\frac{1}{5}} = 2\)<br />- \(x^{\frac{5}{3} \cdot \frac{1}{5}} = x^{\frac{1}{3}}\)<br />- \(y^{\frac{2}{3} \cdot \frac{1}{5}} = y^{\frac{2}{15}}\)<br /><br />Thus, the simplified numerator is:<br />\[<br />2x^{\frac{1}{3}}y^{\frac{2}{15}}<br />\]<br /><br />## Step 2: Simplify the Denominator<br />### Apply the power of a power rule \((a^m)^n = a^{m \cdot n}\) to simplify the denominator.<br />\[<br />(16 x^{-3} y^{2})^{\frac{1}{2}} = 16^{\frac{1}{2}} \cdot x^{-3 \cdot \frac{1}{2}} \cdot y^{2 \cdot \frac{1}{2}}<br />\]<br />Calculate each component:<br />- \(16^{\frac{1}{2}} = 4\)<br />- \(x^{-3 \cdot \frac{1}{2}} = x^{-\frac{3}{2}}\)<br />- \(y^{2 \cdot \frac{1}{2}} = y^{1}\)<br /><br />Thus, the simplified denominator is:<br />\[<br />4x^{-\frac{3}{2}}y<br />\]<br /><br />## Step 3: Divide the Simplified Expressions<br />### Divide the simplified numerator by the simplified denominator.<br />\[<br />\frac{2x^{\frac{1}{3}}y^{\frac{2}{15}}}{4x^{-\frac{3}{2}}y} = \frac{2}{4} \cdot x^{\frac{1}{3} - (-\frac{3}{2})} \cdot y^{\frac{2}{15} - 1}<br />\]<br />Calculate each component:<br />- \(\frac{2}{4} = \frac{1}{2}\)<br />- \(x^{\frac{1}{3} + \frac{3}{2}} = x^{\frac{1}{3} + \frac{9}{6}} = x^{\frac{11}{6}}\)<br />- \(y^{\frac{2}{15} - 1} = y^{\frac{2}{15} - \frac{15}{15}} = y^{-\frac{13}{15}}\)<br /><br />Thus, the final expression is:<br />\[<br />\frac{1}{2} x^{\frac{11}{6}} y^{-\frac{13}{15}}<br />\]<br /><br />## Step 4: Express with Positive Exponents<br />### Rewrite the expression with positive exponents.<br />\[<br />\frac{1}{2} x^{\frac{11}{6}} \cdot \frac{1}{y^{\frac{13}{15}}} = \frac{x^{\frac{11}{6}}}{2y^{\frac{13}{15}}}<br />\]