\( 3 \sqrt[ 1 ] { 54 x } + 4 \sqrt { 75 x } - 6 \sqrt[ 6 ] { 256 x ^ { 2 } } - 8 \sqrt[ 6 ] { 27 x ^ { 3 } } + 5 \sqrt[ 12 ] { 16 x ^ { 4 } } - 2 \sqrt { 108 x } \)

To simplify the expression, let's break it down step by step: 1. **Convert all radicals to their simplest form:** - \(3 \sqrt[1]{54x} = 3(54x) = 162x\) - \(4 \sqrt{75x} = 4 \cdot \sqrt{75} \cdot \sqrt{x} = 4 \cdot 5\sqrt{3} \cdot \sqrt{x} = 20\sqrt{3x}\) - \(6 \sqrt[6]{256x^2} = 6 \cdot \sqrt[6]{256} \cdot \sqrt[6]{x^2} = 6 \cdot 4 \cdot x^{1/3} = 24x^{1/3}\) - \(8 \sqrt[6]{27x^3} = 8 \cdot \sqrt[6]{27} \cdot \sqrt[6]{x^3} = 8 \cdot 3^{1/2} \cdot x^{1/2} = 24\sqrt{3}x^{1/2}\) - \(5 \sqrt[12]{16x^4} = 5 \cdot \sqrt[12]{16} \cdot \sqrt[12]{x^4} = 5 \cdot 2^{4/12} \cdot x^{1/3} = 5 \cdot 2^{1/3} \cdot x^{1/3}\) - \(2 \sqrt{108x} = 2 \cdot \sqrt{108} \cdot \sqrt{x} = 2 \cdot 6\sqrt{3} \cdot \sqrt{x} = 12\sqrt{3x}\) 2. **Substituting back into the expression:** \[ 162x + 20\sqrt{3x} - 24x^{1/3} - 24\sqrt{3}x^{1/2} + 5 \cdot 2^{1/3} x^{1/3} - 12\sqrt{3x} \] 3. **Combine like terms:** - Combine \(20\sqrt{3x}\) and \(-12\sqrt{3x}\) to get \(8\sqrt{3x}\). - Combine \( -24x^{1/3} + 5 \cdot 2^{1/3}x^{1/3} = (-24 + 5 \cdot 2^{1/3})x^{1/3} \). 4. **Final simplified expression:** \[ 162x + 8\sqrt{3x} + (-24 + 5 \cdot 2^{1/3}) x^{1/3} - 24\sqrt{3}x^{1/2} \] This results in a cleaner expression, allowing you to see the components more clearly and work with them in any potential applications or further simplifications!