\( \frac { ( a ^ { \frac { 2 } { 3 } } \cdot b ^ { \frac { 2 } { 2 } } ) ( a ^ { - 1 } , b ^ { 2 } ) } { a ^ { \frac { 1 } { 2 } } \cdot b ^ { \frac { 1 } { 4 } } } \)

To simplify the expression \( \frac { ( a ^ { \frac { 2 } { 3 } } \cdot b ^ { \frac { 2 } { 2 } } ) ( a ^ { - 1 } \cdot b ^ { 2 } ) } { a ^ { \frac { 1 } { 2 } } \cdot b ^ { \frac { 1 } { 4 } } } \), we'll start with the numerator. First, combine the powers of \( a \) and \( b \) in the numerator by applying the product of powers property: 1. For \( a \): - Combine \( a^{\frac{2}{3}} \) and \( a^{-1} \): \[ a^{\frac{2}{3} - 1} = a^{\frac{2}{3} - \frac{3}{3}} = a^{-\frac{1}{3}} \] 2. For \( b \) (noting \( \frac{2}{2} = 1 \)): - Combine \( b^{1} \) and \( b^{2} \): \[ b^{1 + 2} = b^{3} \] After simplifying the numerator, we have: \[ a^{-\frac{1}{3}} \cdot b^{3} \] So the numerator becomes \( a^{-\frac{1}{3}} \cdot b^{3} \). Now the expression is: \[ \frac{a^{-\frac{1}{3}} \cdot b^{3}}{a^{\frac{1}{2}} \cdot b^{\frac{1}{4}}} \] Next, we apply the quotient of powers property: 1. For \( a \): - Combine \( a^{-\frac{1}{3}} \) and \( a^{\frac{1}{2}} \): \[ a^{-\frac{1}{3} - \frac{1}{2}} = a^{-\frac{2}{6} - \frac{3}{6}} = a^{-\frac{5}{6}} \] 2. For \( b \): - Combine \( b^{3} \) and \( b^{\frac{1}{4}} \): \[ b^{3 - \frac{1}{4}} = b^{\frac{12}{4} - \frac{1}{4}} = b^{\frac{11}{4}} \] Thus, we have: \[ \frac{b^{\frac{11}{4}}}{a^{\frac{5}{6}}} \] Putting it all together, the final simplified expression is: \[ \frac{b^{\frac{11}{4}}}{a^{\frac{5}{6}}} \]