Question

Simplify the expression below. Share all work/thinking/calculations to earn full credit. You may want to do the work on paper and then upload an image of your written work rather than try and type your work. \sqrt[4]{ \frac{162x^6}{16x^4} }

Answer 1

`\frac{3x^{(1)/(2)}*\sqrt[4]{2}\text{ }}{2}`

Step-by-step explanation:

`\sqrt[4]{(162x^6)/(16x^4)}`
`\begin{gathered} \sqrt[4]{(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}`
`\begin{gathered} \sqrt[4]{(2x)^4}\text{ = 2x} \\ \sqrt[4]{162x^6}\text{ = (}162x^6)^{(1)/(4)} \\ 162\text{ = 2 }*\text{ 81 = 2 }*3^4 \\ x^6=x^4\text{ }* x^2 \end{gathered}`
`\begin{gathered} \frac{\sqrt[4]{162x^6}}{\sqrt[4]{(2x)^4}}=\text{ }\frac{\sqrt[4]{2*3^4* x^4* x^2}}{2x} \\ =\text{ }\frac{3* x*\sqrt[4]{2* x^2}}{2x} \\ =\text{ }\frac{3x*\sqrt[4]{2* x^2}}{2x} \end{gathered}`
`\begin{gathered} \frac{3*\sqrt[4]{2x^2}}{2}\text{ = }\frac{3*\sqrt[4]{2}\text{ }*\sqrt[4]{x^2}}{2} \\ \sqrt[4]{x^2}\text{ = (}x^2)^{(1)/(4)}\text{ = }x^{(2)/(4)}\text{ = }x^{(1)/(2)} \\ \frac{3*\sqrt[4]{2}\text{ }*\sqrt[4]{x^2}}{2}\text{=}\frac{3x^{(1)/(2)}*\sqrt[4]{2}\text{ }}{2} \\ \\ \frac{3*\sqrt[4]{2x^2}}{2}\text{ or }\frac{3x^{(1)/(2)}*\sqrt[4]{2}\text{ }}{2} \end{gathered}`