implify the expression. Assume all variables are positiv \( \begin{array}{lll}\text { 3. } \sqrt[3]{27 q^{9}} & \text { 14. } \sqrt[5]{\frac{x^{10}}{y^{5}}} & \text { 15. } \frac{6 x y^{3 / 4}}{3 x^{1 / 2} y^{1 / 2}}\end{array} \)
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For the first expression, \( \sqrt[3]{27 q^{9}} \), we can simplify it by recognizing that \( 27 = 3^3 \), so \( \sqrt[3]{27} = 3 \). For \( q^{9} \), since the index of the root is 3, we take \( q^{9/3} = q^{3} \). Thus, the simplified expression is \( 3q^{3} \). Moving on to the second expression, \( \sqrt[5]{\frac{x^{10}}{y^{5}}} \), we can split it into two parts: \( \sqrt[5]{x^{10}} \) and \( \sqrt[5]{y^{5}} \). The first part simplifies to \( x^{10/5} = x^{2} \) and the second part simplifies to \( y^{5/5} = y^{1} \). Therefore, the expression simplifies to \( \frac{x^{2}}{y} \). Finally, for the third expression \( \frac{6 x y^{3/4}}{3 x^{1/2} y^{1/2}} \), we can simplify the coefficients to get \( \frac{6}{3} = 2 \). For the variable \( x \), \( \frac{x}{x^{1/2}} = x^{1/2} \). For \( y \), we apply the rule of exponents: \( \frac{y^{3/4}}{y^{1/2}} = y^{(3/4 - 2/4)} = y^{1/4} \). Thus, the final simplified expression is \( 2 x^{1/2} y^{1/4} \).
