To determine which expression is equivalent to [tex]\( x^{-\frac{5}{3}} \)[/tex], let's simplify and analyze each option step-by-step.
### Step-by-Step Analysis
#### 1. Analyze [tex]\( x^{-\frac{5}{3}} \)[/tex]
This can be written as:
[tex]\[ x^{-\frac{5}{3}} = \frac{1}{x^{\frac{5}{3}}} \][/tex]
#### 2. Equivalent expressions:
We need to examine the given options to see which one matches the above simplification.
#### 3. Analyze Option 1: [tex]\( \frac{1}{\sqrt[5]{x^3}} \)[/tex]
[tex]\[
\frac{1}{\sqrt[5]{x^3}} = \frac{1}{x^{\frac{3}{5}}}
\][/tex]
Since [tex]\( \frac{3}{5} \neq \frac{5}{3} \)[/tex], this option is not equivalent to [tex]\( x^{-\frac{5}{3}} \)[/tex].
#### 4. Analyze Option 2: [tex]\( \frac{1}{\sqrt[3]{x^5}} \)[/tex]
[tex]\[
\frac{1}{\sqrt[3]{x^5}} = \frac{1}{x^{\frac{5}{3}}}
\][/tex]
This matches [tex]\( \frac{1}{x^{\frac{5}{3}}} \)[/tex], so this option is a potential equivalent.
#### 5. Analyze Option 3: [tex]\( -\sqrt[3]{x^5} \)[/tex]
[tex]\[
-\sqrt[3]{x^5} = -x^{\frac{5}{3}}
\][/tex]
This does not include the negative exponent, so it is not equivalent to [tex]\( x^{-\frac{5}{3}} \)[/tex].
#### 6. Analyze Option 4: [tex]\( -\sqrt[5]{x^3} \)[/tex]
[tex]\[
-\sqrt[5]{x^3} = -x^{\frac{3}{5}}
\][/tex]
Again, since [tex]\( \frac{3}{5} \neq \frac{5}{3} \)[/tex], and the expression should not have a negative sign, this option is not equivalent to [tex]\( x^{-\frac{5}{3}} \)[/tex].
### Conclusion
After analyzing each option, we find that the only expression equivalent to [tex]\( x^{-\frac{5}{3}} \)[/tex] is:
[tex]\[
\frac{1}{\sqrt[3]{x^5}}
\][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{1}{\sqrt[3]{x^5}}} \][/tex]
