To calculate: \sqrt{\frac{x}{8}} The simplified value of the radical expression.

Step 1
Formula used:
The relation between radical and rational exponent notation is expressed as,
$\sqrt[n]{a}=a^{1/n}$
Here, a is the radicand, and n is the index of the radical.
$(\sqrt[n]{a}{)}^m=\sqrt[n]{a^m}$
${\displaystyle =a^{\frac{m}{n}}}$
The product rule for radicals,
$\sqrt[n]{ab}-\sqrt[n]{a}\cdot \sqrt[n]{b}$
The quotient rule for radicals,
${\displaystyle \sqrt{n}\left\{\frac{a}{b}\right\}=\frac{\sqrt{n}\left\{a\right\}}{\sqrt{n}\left\{b\right\}}}$
Step 2
Consider the given expression,
${\displaystyle \sqrt{\frac{x}{8}}}$
Use the relation between radical and rational exponent notation to simplify the expression,
${\displaystyle \sqrt{\frac{x}{8}}={\left(\frac{x}{8}\right)}^{\frac{1}{2}}}$
${\displaystyle ={\left(\frac{x}{4\cdot 2}\right)}^{\frac{1}{2}}}$
${\displaystyle =\frac{x^{\frac{1}{2}}}{{\left(4\right)}^{\frac{1}{2}}\cdot {\left(2\right)}^{\frac{1}{2}}}}$
${\displaystyle =\frac{x^{\frac{1}{2}}}{{\left(2^2\right)}^{\frac{1}{2}}\cdot {\left(2\right)}^{\frac{1}{2}}}}$
Further simplify by using the product rule and quotient rule for radicals,
${\displaystyle \sqrt{\frac{x}{8}}=\frac{\sqrt{x}}{2\sqrt{2}}}$
${\displaystyle =\frac{\sqrt{x}}{2\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}}$
${\displaystyle =\frac{\sqrt{x}\cdot \sqrt{2}}{2\cdot \sqrt{2}\cdot \sqrt{2}}}$
${\displaystyle =\frac{\sqrt{2x}}{4}}$
Hence, the simplified value of the provided expression is ${\displaystyle \frac{\sqrt{2x}}{4}}$