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 the 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 the fractional exponent rule

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

Step3: Combine the two 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:

B. \(\frac{1}{\sqrt[3]{x^{5}}}\) (assuming the second option is labeled B, with the expression \(\frac{1}{\sqrt[3]{x^{5}}}\))