What is the simplified form of the following expression? \( 7(\sqrt[3]{2 x})-3(\sqrt[3]{16 x})-3(\sqrt[3]{8 x}) \)

To simplify the expression \( 7(\sqrt[3]{2 x})-3(\sqrt[3]{16 x})-3(\sqrt[3]{8 x}) \), we first express terms under the cube root in a convenient way. 1. Notice that \( \sqrt[3]{16x} = \sqrt[3]{16}\cdot\sqrt[3]{x} = \sqrt[3]{(2^4)}\cdot\sqrt[3]{x} = 2^{4/3}\cdot\sqrt[3]{x} \). 2. Also, \( \sqrt[3]{8x} = \sqrt[3]{8}\cdot\sqrt[3]{x} = 2^{3/3}\cdot\sqrt[3]{x} = 2\cdot\sqrt[3]{x} \). Now, rewrite the expression: \[ 7\sqrt[3]{2x} - 3\cdot2^{4/3}\cdot\sqrt[3]{x} - 3\cdot2\cdot\sqrt[3]{x} \] Now, let's factor out \( \sqrt[3]{x} \) from the last two terms: \[ = 7\sqrt[3]{2x} - 3\left(2^{4/3} + 2\right)\sqrt[3]{x} \] Next, we calculate the combination \( 2^{4/3} + 2 \): Since \( 2 = 2^{3/3} \), we can add these: \[ 2^{4/3} + 2^{3/3} = \frac{2^4 + 2^3}{2^{1/3}} = \frac{16 + 8}{2^{1/3}} = \frac{24}{2^{1/3}} \] However, returning to our simplified form, we may also express everything in terms of \( \sqrt[3]{x} \): \[ = 7\sqrt[3]{2}\cdot\sqrt[3]{x} - 3\left(2^{4/3}+2\right)\sqrt[3]{x} \] This makes it clear how we might group terms together. Factoring out \( \sqrt[3]{x} \): \[ = \sqrt[3]{x}\left(7\sqrt[3]{2} - 3(2^{4/3} + 2)\right) \] Now we will leave it like this since \( 2^{4/3} + 2 \) can't be simplified much further without context. So the final answer is: \[ \sqrt[3]{x}\left(7\sqrt[3]{2} - 3\left(2^{4/3} + 2\right)\right) \]