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Rewrite the expression in the form $x^{n}$ . $\newline$ Write the exponent as an integer, fraction, or an exact decimal (not a mixed number). $\newline$ $\sqrt[4]{\frac{x^{2}}{x^{\frac{2}{3}}}}=\square$

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Roots of rational numbers

Full solution

Q. Rewrite the expression in the form

x^{n}

.

\newline

Write the exponent as an integer, fraction, or an exact decimal (not a mixed number).

\newline

\sqrt[4]{\frac{x^{2}}{x^{\frac{2}{3}}}}=\square

Apply Exponent Property: Apply the property of exponents that states $(x^{a})/(x^{b}) = x^{a-b}$ to the expression inside the fourth root. $\sqrt[4]{\left(\frac{x^{2}}{x^{\frac{2}{3}}}\right)} = \sqrt[4]{x^{2 - \frac{2}{3}}}$
Subtract Exponents: Subtract the exponents: $2 - \left(\frac{2}{3}\right) = \left(\frac{6}{3}\right) - \left(\frac{2}{3}\right) = \frac{4}{3}$ . $\sqrt[4]{x^{2 - \left(\frac{2}{3}\right)}} = \sqrt[4]{x^{\frac{4}{3}}}$
Apply Root Property: Apply the property of exponents that states $\sqrt[n]{x^{m}} = x^{\frac{m}{n}}$ to the expression. $\sqrt[4]{x^{\frac{4}{3}}} = x^{\left(\frac{4}{3}\right)/4}$
Simplify Exponent: Simplify the exponent by dividing $\frac{4}{3}$ by $4$ , which is the same as multiplying $\frac{4}{3}$ by $\frac{1}{4}$ . $\newline$ $x^{\left(\frac{4}{3}\right)/4} = x^{\left(\frac{4}{3}\right)\cdot\left(\frac{1}{4}\right)} = x^{\frac{1}{3}}$

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