[tex]\frac{3x^{\frac{1}{2}}\times\sqrt[4]{2}\text{ }}{2}[/tex]
Explanation:
[tex]\sqrt[4]{\frac{162x^6}{16x^4}}[/tex][tex]\begin{gathered} \sqrt[4]{\frac{162x^6}{16x^4}}\text{ = }\frac{\sqrt[4]{162x^6}}{\sqrt[4]{16x^4}} \\ 16x^4=2^4x^4=(2x)^4 \\ \frac{\sqrt[4]{162x^6}}{\sqrt[4]{16x^4}}\text{ = }\frac{\sqrt[4]{162x^6}}{\sqrt[4]{(2x)^4}} \end{gathered}[/tex][tex]\begin{gathered} \sqrt[4]{(2x)^4}\text{ = 2x} \\ \sqrt[4]{162x^6}\text{ = (}162x^6)^{\frac{1}{4}} \\ 162\text{ = 2 }\times\text{ 81 = 2 }\times3^4 \\ x^6=x^4\text{ }\times x^2 \end{gathered}[/tex][tex]\begin{gathered} \frac{\sqrt[4]{162x^6}}{\sqrt[4]{(2x)^4}}=\text{ }\frac{\sqrt[4]{2\times3^4\times x^4\times x^2}}{2x} \\ =\text{ }\frac{3\times x\times\sqrt[4]{2\times x^2}}{2x} \\ =\text{ }\frac{3x\times\sqrt[4]{2\times x^2}}{2x} \end{gathered}[/tex][tex]\begin{gathered} \frac{3\times\sqrt[4]{2x^2}}{2}\text{ = }\frac{3\times\sqrt[4]{2}\text{ }\times\sqrt[4]{x^2}}{2} \\ \sqrt[4]{x^2}\text{ = (}x^2)^{\frac{1}{4}}\text{ = }x^{\frac{2}{4}}\text{ = }x^{\frac{1}{2}} \\ \frac{3\times\sqrt[4]{2}\text{ }\times\sqrt[4]{x^2}}{2}\text{=}\frac{3x^{\frac{1}{2}}\times\sqrt[4]{2}\text{ }}{2} \\ \\ \frac{3\times\sqrt[4]{2x^2}}{2}\text{ or }\frac{3x^{\frac{1}{2}}\times\sqrt[4]{2}\text{ }}{2} \end{gathered}[/tex]
