Question

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

Explanation:

Step1: Recall exponent rules

Recall that \(a^{-n}=\frac{1}{a^n}\) and \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\). So for \(x^{-\frac{5}{3}}\), first apply the negative exponent rule: \(x^{-\frac{5}{3}}=\frac{1}{x^{\frac{5}{3}}}\).

Step2: Apply fractional exponent rule

Now, \(x^{\frac{5}{3}}=\sqrt[3]{x^5}\) (since \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\) with \(a = x\), \(m = 5\), \(n = 3\)). So \(\frac{1}{x^{\frac{5}{3}}}=\frac{1}{\sqrt[3]{x^5}}\). Wait, but let's check the options again. Wait, maybe I misread the original exponent. Wait, the original problem is "which expression is equivalent to \(x^{-\frac{5}{3}}\)?" Wait, looking at the options: the third option is \(\frac{1}{\sqrt[3]{x^5}}\)? Wait, no, the options are: first \(\frac{1}{\sqrt[5]{x^3}}\), second \(\frac{1}{\sqrt[3]{x^5}}\), third \(-\sqrt[3]{x^5}\), fourth \(-\sqrt[5]{x^3}\). Wait, no, the user's image: the options are (from left to right? Wait, the text is "Which expression is equivalent to \(x^{-\frac{5}{3}}\)?", and the options are: first \(\frac{1}{\sqrt[5]{x^3}}\), second \(\frac{1}{\sqrt[3]{x^5}}\), third \(-\sqrt[3]{x^5}\), fourth \(-\sqrt[5]{x^3}\). Wait, let's redo:

Using exponent rules: \(a^{-b}=\frac{1}{a^b}\), so \(x^{-\frac{5}{3}}=\frac{1}{x^{\frac{5}{3}}}\). Then, \(x^{\frac{5}{3}}=\sqrt[3]{x^5}\) (because \(x^{\frac{m}{n}}=\sqrt[n]{x^m}\), here \(m = 5\), \(n = 3\)). So \(\frac{1}{x^{\frac{5}{3}}}=\frac{1}{\sqrt[3]{x^5}}\), which is the second option? Wait, no, the second option in the image (from the left, the second circle) is \(\frac{1}{\sqrt[3]{x^5}}\)? Wait, the user's image: the options are:

1. \(\frac{1}{\sqrt[5]{x^3}}\)

1. \(\frac{1}{\sqrt[3]{x^5}}\)

1. \(-\sqrt[3]{x^5}\)

1. \(-\sqrt[5]{x^3}\)

So applying \(x^{-\frac{5}{3}}=\frac{1}{x^{\frac{5}{3}}}=\frac{1}{\sqrt[3]{x^5}}\) (since \(x^{\frac{5}{3}}=\sqrt[3]{x^5}\)), so the correct option is the second one, which is \(\frac{1}{\sqrt[3]{x^5}}\). Wait, but let's confirm:

\(x^{-\frac{5}{3}}=\frac{1}{x^{\frac{5}{3}}}\), and \(x^{\frac{5}{3}}=\sqrt[3]{x^5}\) (by \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\)), so \(\frac{1}{x^{\frac{5}{3}}}=\frac{1}{\sqrt[3]{x^5}}\), which is the second option.

Answer:

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