 # To calculate: \sqrt{\frac{x}{8}} The simplified value of the radical expression. Ernstfalld 2021-08-06 Answered
To calculate: $\sqrt{\frac{x}{8}}$
The simplified value of the radical expression.
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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.
$\left(\sqrt[n]{a}{\right)}^{m}=\sqrt[n]{{a}^{m}}$
$={a}^{\frac{m}{n}}$
$\sqrt[n]{ab}-\sqrt[n]{a}\cdot \sqrt[n]{b}$
$\sqrt{n}\left\{\frac{a}{b}\right\}=\frac{\sqrt{n}\left\{a\right\}}{\sqrt{n}\left\{b\right\}}$
Step 2
Consider the given expression,
$\sqrt{\frac{x}{8}}$
Use the relation between radical and rational exponent notation to simplify the expression,
$\sqrt{\frac{x}{8}}={\left(\frac{x}{8}\right)}^{\frac{1}{2}}$
$={\left(\frac{x}{4\cdot 2}\right)}^{\frac{1}{2}}$
$=\frac{{x}^{\frac{1}{2}}}{{\left(4\right)}^{\frac{1}{2}}\cdot {\left(2\right)}^{\frac{1}{2}}}$
$=\frac{{x}^{\frac{1}{2}}}{{\left({2}^{2}\right)}^{\frac{1}{2}}\cdot {\left(2\right)}^{\frac{1}{2}}}$
Further simplify by using the product rule and quotient rule for radicals,
$\sqrt{\frac{x}{8}}=\frac{\sqrt{x}}{2\sqrt{2}}$
$=\frac{\sqrt{x}}{2\sqrt{2}}×\frac{\sqrt{2}}{\sqrt{2}}$
$=\frac{\sqrt{x}\cdot \sqrt{2}}{2\cdot \sqrt{2}\cdot \sqrt{2}}$
$=\frac{\sqrt{2x}}{4}$
Hence, the simplified value of the provided expression is $\frac{\sqrt{2x}}{4}$

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