Question

which expression is equivalent to ( x^{-\frac{5}{3}} )?
( circ \frac{1}{sqrt5{x^3}} )
( circ \frac{1}{sqrt3{x^5}} )
( circ -sqrt3{x^5} )
( circ -sqrt5{x^3} )

Explanation:

Step1: Recall Negative Exponent Rule

The negative exponent rule states that \( a^{-n} = \frac{1}{a^n} \), so \( x^{-\frac{5}{3}}=\frac{1}{x^{\frac{5}{3}}} \).

Step2: Recall Rational Exponent to Radical Rule

The rule for converting a rational exponent to a radical is \( a^{\frac{m}{n}}=\sqrt[n]{a^m} \), so \( x^{\frac{5}{3}}=\sqrt[3]{x^5} \).

Step3: Combine the Rules

Substituting \( x^{\frac{5}{3}}=\sqrt[3]{x^5} \) into \( \frac{1}{x^{\frac{5}{3}}} \), we get \( \frac{1}{\sqrt[3]{x^5}} \).

Answer:

\(\boldsymbol{\frac{1}{\sqrt[3]{x^5}}}\) (the second option)