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

kuCAu 2021-02-12 Answered
Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
\(\frac{\sqrt[3]{8x^{2}}}{\sqrt{x}}\)

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Expert Answer

Tasneem Almond
Answered 2021-02-13 Author has 19702 answers
Given:
\(\frac{\sqrt[3]{8x^{2}}}{\sqrt{x}}\)
“For any rational exponent min in lowest terms, where m and n are integers and \(n\ >\ 0,\) we define
\(a^{^m/_n}=(\sqrt[n]{a})^{m}=\sqrt[n]{a^{m}}\)
If nis even, then we require that \(a\ \geq\ 0”\)
Calculation:
Consider the given expression,
\(\frac{\sqrt[3]{8x^{2}}}{\sqrt{x}}\)
Apply radical rule : \(\sqrt[n]{ab}=\sqrt[n]{a}\ \sqrt[n]{b}\) in the numerator, we get
\(\sqrt[3]{8x^{2}}=\sqrt[3]{8}\ \cdot\ \sqrt[3]{x^{2}}\)
\(=\sqrt[3]{2^{3}}\ \cdot\ \sqrt[3]{x^{2}}\)
\(=2\sqrt[3]{x^{2}}\)
Substitute \(\frac{\sqrt[3]{8x^{2}}}{\sqrt{x}}=2\sqrt[3]{x^{2}}\) we get
By using the law of exponents,
\(\frac{2\sqrt[3]{x^{2}}}{x}=\frac{2x^{\frac{2}{3}}}{x^{\frac{1}{2}}}\)
Apply exponent rule: \(\frac{x^{a}}{x^{b}}=x^{a\ -\ b}\) we get
\(\frac{2x^{\frac{2}{3}}}{x^{\frac{1}{3}}}=2x^{\frac{2}{3}\ -\ \frac{1}{2}}\)
\(= 2x^{\frac{1}{6}}\)
Final statement:
The simplyfied form of \(\frac{\sqrt[3]{8x^{2}}}{\sqrt{x}} is 2x^{\frac{1}{6}}\)
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