We are given the following radical expression.
[tex]\sqrt{\frac{2}{x}}\cdot\sqrt{\frac{x^2}{8}}[/tex]
Let us simplify the above expression.
Recall the product rule of radicals given by
[tex]\sqrt{a}\cdot\sqrt{b}=\sqrt{a\cdot b}[/tex]
Apply the above rule to the given expression.
[tex]\sqrt{\frac{2}{x}}\cdot\sqrt{\frac{x^2}{8}}\Rightarrow\sqrt{\frac{2x^2}{8x}}\Rightarrow\sqrt{\frac{x}{4}}[/tex]
Recall the quotient rule of radicals given by
[tex]\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}[/tex]
Apply the above rule to the given expression.
[tex]\sqrt{\frac{x}{4}}\Rightarrow\frac{\sqrt{x}}{\sqrt{4}}\Rightarrow\frac{\sqrt{x}}{2}[/tex]
Therefore, the simplified expression is
[tex]\frac{\sqrt{x}}{2}[/tex]
Option A is the correct answer.
