Given that \( (\sqrt[3]{4})^{k}=a \), find the values of the following in terms of \( a \). \( \begin{array}{lll}\text { (a) } 4^{k} & \text { (b) } 16^{\frac{k}{3}} & \text { (c) } 2^{k+1}\end{array} \)

Given that \( (\sqrt[3]{4})^{k} = a \), we can express \( \sqrt[3]{4} \) as \( 4^{1/3} \). Therefore, we can rewrite the equation as: \[ (4^{1/3})^{k} = 4^{k/3} = a. \] Now, we can find the values of \( 4^{k} \), \( 16^{\frac{k}{3}} \), and \( 2^{k+1} \) in terms of \( a \). **(a)** To find \( 4^{k} \): Since \( a = 4^{k/3} \), we can express \( 4^{k} \) in terms of \( a \) by raising both sides to the power of \( 3 \): \[ 4^{k} = (4^{k/3})^3 = a^3. \] **(b)** To find \( 16^{\frac{k}{3}} \): We know that \( 16 \) can be expressed as \( 4^2 \). Hence, we can rewrite \( 16^{\frac{k}{3}} \) as: \[ 16^{\frac{k}{3}} = (4^2)^{\frac{k}{3}} = 4^{\frac{2k}{3}}. \] Using the fact that \( 4^{k/3} = a \), we have: \[ 4^{\frac{2k}{3}} = (4^{k/3})^2 = a^2. \] **(c)** To find \( 2^{k+1} \): We start by rewriting \( 4 \) as \( 2^2 \). Thus, we can express \( 4^{k/3} \): \[ 4^{k/3} = (2^2)^{k/3} = 2^{\frac{2k}{3}}. \] Now, to find \( 2^{k+1} \): \[ 2^{k+1} = 2^k \cdot 2 = 2^{k} \cdot 2. \] From earlier, we know \( 4^{k} = 2^{2k} \). To relate it to \( a \): \[ 4^{k/3} = a \implies 4^{k} = a^3 \implies 2^{2k} = a^3 \implies 2^{k} = a^{3/2}. \] Therefore: \[ 2^{k+1} = 2 \cdot 2^{k} = 2 \cdot a^{3/2}. \] In summary, we can express the final results as follows: (a) \( 4^k = a^3 \) (b) \( 16^{\frac{k}{3}} = a^2 \) (c) \( 2^{k+1} = 2a^{\frac{3}{2}} \)