Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. frac{sqrt[3]{8x^{2}}}{sqrt{x}}

Question
Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
\(\displaystyle{\frac{{\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}\)

Answers (1)

2021-03-06
Step 1
\(\displaystyle{\frac{{\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}\)
Law of exponents:
\(\displaystyle{a}^{{\frac{{m}}{{n}}}}={\left(\sqrt{{{n}}}{\left\lbrace{a}\right\rbrace}\right)}^{{{m}}}=\sqrt{{{n}}}{\left\lbrace{a}^{{{m}}}\right\rbrace}\)
If n is even, then we require that \(\displaystyle{a}\geq{0}\)
Step 2
Consider the given expression,
1) \(\displaystyle{\frac{{\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}\)
Apply radical rule: \(\displaystyle\sqrt{{{n}}}{\left\lbrace{a}{b}\right\rbrace}=\sqrt{{{n}}}{\left\lbrace{a}\right\rbrace}\sqrt{{{n}}}{\left\lbrace{b}\right\rbrace}\) in the numerator, we get
\(\displaystyle\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}=\sqrt{{{3}}}{\left\lbrace{8}\right\rbrace}\cdot\sqrt{{{3}}}{\left\lbrace{x}^{{{2}}}\right\rbrace}\)
\(\displaystyle=\sqrt{{{3}}}{\left\lbrace{2}^{{{3}}}\right\rbrace}\cdot\sqrt{{{3}}}{\left\lbrace{x}^{{{2}}}\right\rbrace}\)
\(\displaystyle={2}\sqrt{{{3}}}{\left\lbrace{x}^{{{2}}}\right\rbrace}\)
Substitute \(\displaystyle\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}={2}\sqrt{{{3}}}{\left\lbrace{x}^{{{2}}}\right\rbrace}\) in equation (1), we get
\(\displaystyle{\frac{{{2}\sqrt{{{3}}}{\left\lbrace{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}={\frac{{{2}{x}^{{{\frac{{{2}}}{{{3}}}}}}}}{{{x}^{{{\frac{{{1}}}{{{2}}}}}}}}}\)
Apply exponent rule: \(\displaystyle{\frac{{{x}^{{{a}}}}}{{{x}^{{{b}}}}}}={x}^{{{a}-{b}}}\), we get
\(\displaystyle{\frac{{{2}{x}^{{{\frac{{{2}}}{{{3}}}}}}}}{{{x}^{{{\frac{{{1}}}{{{2}}}}}}}}}={2}{x}^{{{\frac{{{2}}}{{{3}}}}-{\frac{{{1}}}{{{2}}}}}}\)
\(\displaystyle={2}{x}^{{{\frac{{{1}}}{{{6}}}}}}\)
The simplified form of \(\displaystyle{\frac{{\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}\) is \(\displaystyle{2}{x}^{{{\frac{{{1}}}{{{6}}}}}}\)
0

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