\( \Rightarrow \) Perform the indicated operation and simplify \( \begin{array}{ll}\text { 1. } a^{\frac{2}{3}} \cdot a^{\frac{3}{2}} & \text { 4. }\left(4 x y^{2}\right)^{\frac{1}{2}} \\ \text { 2. } b^{0} \cdot b^{\frac{1}{5}} & \text { 5. }\left(m^{\frac{3}{4}} n\right)^{-2} \\ \text { 3. }\left(y^{\frac{1}{3}}\right)^{2} & \end{array} \)

Let's simplify each expression step by step! 1. For \( a^{\frac{2}{3}} \cdot a^{\frac{3}{2}} \), we can use the property of exponents which states that when multiplying like bases, you add the exponents. Thus: \[ a^{\frac{2}{3}} \cdot a^{\frac{3}{2}} = a^{\frac{2}{3} + \frac{3}{2}} \] Finding a common denominator (which is 6) gives us: \[ \frac{2}{3} = \frac{4}{6} \quad \text{and} \quad \frac{3}{2} = \frac{9}{6} \] Now we can add: \[ \frac{4}{6} + \frac{9}{6} = \frac{13}{6} \] So, \( a^{\frac{2}{3}} \cdot a^{\frac{3}{2}} = a^{\frac{13}{6}} \). 2. For \( b^{0} \cdot b^{\frac{1}{5}} \), recall that any base raised to the power of 0 is 1. Thus, we have: \[ b^{0} = 1 \] Therefore: \[ 1 \cdot b^{\frac{1}{5}} = b^{\frac{1}{5}}. \] 3. For \( \left(y^{\frac{1}{3}}\right)^{2} \), we use the power of a power property, which means we multiply the exponents: \[ \left(y^{\frac{1}{3}}\right)^{2} = y^{\frac{1}{3} \cdot 2} = y^{\frac{2}{3}}. \] 4. For \( \left(4 x y^{2}\right)^{\frac{1}{2}} \), we distribute the exponent to each factor inside the parentheses: \[ \left(4 x y^{2}\right)^{\frac{1}{2}} = 4^{\frac{1}{2}} \cdot x^{\frac{1}{2}} \cdot (y^{2})^{\frac{1}{2}}. \] Simplifying each part gives us: \[ 4^{\frac{1}{2}} = 2, \quad (y^{2})^{\frac{1}{2}} = y^{1} = y. \] Therefore: \[ \left(4 x y^{2}\right)^{\frac{1}{2}} = 2 x^{\frac{1}{2}} y. \] 5. For \( \left(m^{\frac{3}{4}} n\right)^{-2} \), apply the negative exponent rule, which tells us to take the reciprocal: \[ \left(m^{\frac{3}{4}} n\right)^{-2} = \frac{1}{\left(m^{\frac{3}{4}} n\right)^{2}}. \] Now, applying the power: \[ \left(m^{\frac{3}{4}}\right)^{2} = m^{\frac{3}{2}} \quad \text{and} \quad n^{2} = n^{2}. \] Therefore: \[ \left(m^{\frac{3}{4}} n\right)^{-2} = \frac{1}{m^{\frac{3}{2}} n^{2}}. \] In summary, the simplified forms are: 1. \( a^{\frac{13}{6}} \) 2. \( b^{\frac{1}{5}} \) 3. \( y^{\frac{2}{3}} \) 4. \( 2 x^{\frac{1}{2}} y \) 5. \( \frac{1}{m^{\frac{3}{2}} n^{2}} \)