plest form. Assume all variables are positive. \( \begin{array}{ll}\text { b. } \frac{x}{\sqrt[3]{y^{8}}} & \text { c. } \frac{14 x y^{1 / 3}}{2 x^{3 / 4} z^{-6}}\end{array} \)

To simplify \( \frac{x}{\sqrt[3]{y^{8}}} \), recall that \( \sqrt[3]{y^{8}} = y^{8/3} \). Thus, we can rewrite the expression as: \[ \frac{x}{y^{8/3}} = xy^{-8/3} \] So the expression in its simplest form is \( xy^{-8/3} \). For \( \frac{14 x y^{1 / 3}}{2 x^{3 / 4} z^{-6}} \), we can simplify it step-by-step. First, divide the coefficients: \[ \frac{14}{2} = 7 \] Next, simplify the \( x \) terms by applying the properties of exponents: \[ \frac{x}{x^{3/4}} = x^{1 - 3/4} = x^{1/4} \] Now for \( z^{-6} \) in the denominator, we can move it to the numerator as \( z^{6} \). This gives us: \[ \frac{y^{1/3}}{1} = y^{1/3} \] So, combining all this together, we find: \[ \frac{14 x y^{1 / 3}}{2 x^{3 / 4} z^{-6}} = 7 x^{1/4} y^{1/3} z^{6} \] The final simplified form is \( 7 x^{1/4} y^{1/3} z^{6} \).