To determine which given expression is equivalent to [tex]\( x^{-\frac{5}{3}} \)[/tex], let's rewrite [tex]\( x^{-\frac{5}{3}} \)[/tex] using properties of exponents and roots.
1. Recall the property of negative exponents: [tex]\( x^{-a} = \frac{1}{x^a} \)[/tex].
Therefore, [tex]\( x^{-\frac{5}{3}} = \frac{1}{x^{\frac{5}{3}}} \)[/tex].
2. We can further rewrite [tex]\( x^{\frac{5}{3}} \)[/tex] using the property of fractional exponents in terms of roots: [tex]\( x^{\frac{a}{b}} = \sqrt[b]{x^a} \)[/tex].
Thus, [tex]\( x^{\frac{5}{3}} = \sqrt[3]{x^5} \)[/tex].
3. Substituting this back into our expression:
[tex]\[
x^{-\frac{5}{3}} = \frac{1}{x^{\frac{5}{3}}} = \frac{1}{\sqrt[3]{x^5}}
\][/tex]
Now let's compare this to the given choices:
1. [tex]\(\frac{1}{\sqrt[5]{x^3}}\)[/tex]: This expression corresponds to [tex]\( x^{-\frac{3}{5}} \)[/tex] because [tex]\( \sqrt[5]{x^3} = x^{\frac{3}{5}} \)[/tex]. Hence, [tex]\( \frac{1}{\sqrt[5]{x^3}} = x^{-\frac{3}{5}} \)[/tex], which is not equivalent to [tex]\( x^{-\frac{5}{3}} \)[/tex].
2. [tex]\(\frac{1}{\sqrt[3]{x^5}}\)[/tex]: This matches our transformed expression [tex]\( x^{-\frac{5}{3}} = \frac{1}{\sqrt[3]{x^5}} \)[/tex].
3. [tex]\(-\sqrt[3]{x^5}\)[/tex]: This represents the negative of the cube root of [tex]\( x^5 \)[/tex], not related to the expression [tex]\( x^{-\frac{5}{3}} \)[/tex].
4. [tex]\(-\sqrt[5]{x^3}\)[/tex]: This represents the negative of the fifth root of [tex]\( x^3 \)[/tex], also not related to the expression [tex]\( x^{-\frac{5}{3}} \)[/tex].
Therefore, the expression that is equivalent to [tex]\( x^{-\frac{5}{3}} \)[/tex] is:
[tex]\[
\boxed{\frac{1}{\sqrt[3]{x^5}}}
\][/tex]
