Question: b) \[ 3m^{\frac{1}{4}} \left( mn^{\frac{1}{3}} \right) = \]

b)

\[ 3m^{\frac{1}{4}} \left( mn^{\frac{1}{3}} \right) = \]

Solution

To solve the given expression \(3m^{\frac{1}{4}} \left( mn^{\frac{1}{3}} \right)^{\frac{3}{4}}\), follow these steps: First, apply the power of a power rule inside the parentheses: \[ (mn^{\frac{1}{3}})^{\frac{3}{4}} = m^{\frac{3}{4}} \cdot (n^{\frac{1}{3}})^{\frac{3}{4}} \] Now simplify the expression further: \[ (mn^{\frac{1}{3}})^{\frac{3}{4}} = m^{\frac{3}{4}} \cdot n^{\frac{1}{4}} \] Next, multiply by \(3m^{\frac{1}{4}}\): \[ 3m^{\frac{1}{4}} \cdot m^{\frac{3}{4}} \cdot n^{\frac{1}{4}} \] Combine the \(m\) terms: \[ 3m^{\frac{1}{4} + \frac{3}{4}} \cdot n^{\frac{1}{4}} = 3m^{1} \cdot n^{\frac{1}{4}} \] Simplify to get the final expression: \[ 3m \cdot n^{\frac{1}{4}} \] Therefore, the simplified expression is \(3mn^{\frac{1}{4}}\).