Answer :
To determine which expression is equivalent to \( x^{-\frac{5}{3}} \), we need to simplify this expression step-by-step.
1. Understanding Negative Exponents:
[tex]\[ x^{-\frac{5}{3}} = \frac{1}{x^{\frac{5}{3}}} \][/tex]
A negative exponent indicates that we take the reciprocal of the base with the positive of that exponent.
2. Converting \( x^{\frac{5}{3}} \) to Radical Form:
We can express \( x^{\frac{5}{3}} \) using roots.
[tex]\[ x^{\frac{5}{3}} = (x^5)^{\frac{1}{3}} = \sqrt[3]{x^5} \][/tex]
This means that raising \( x^5 \) to the power of \(\frac{1}{3}\) is equivalent to finding the cube root of \( x^5 \).
3. Simplifying the Expression:
Substituting back in, we have:
[tex]\[ x^{-\frac{5}{3}} = \frac{1}{x^{\frac{5}{3}}} = \frac{1}{\sqrt[3]{x^5}} \][/tex]
Therefore, the expression equivalent to \( x^{-\frac{5}{3}} \) is:
[tex]\[ \frac{1}{\sqrt[3]{x^5}} \][/tex]
Among the given options, this corresponds to:
[tex]\[ \boxed{\frac{1}{\sqrt[3]{x^5}}} \][/tex]
1. Understanding Negative Exponents:
[tex]\[ x^{-\frac{5}{3}} = \frac{1}{x^{\frac{5}{3}}} \][/tex]
A negative exponent indicates that we take the reciprocal of the base with the positive of that exponent.
2. Converting \( x^{\frac{5}{3}} \) to Radical Form:
We can express \( x^{\frac{5}{3}} \) using roots.
[tex]\[ x^{\frac{5}{3}} = (x^5)^{\frac{1}{3}} = \sqrt[3]{x^5} \][/tex]
This means that raising \( x^5 \) to the power of \(\frac{1}{3}\) is equivalent to finding the cube root of \( x^5 \).
3. Simplifying the Expression:
Substituting back in, we have:
[tex]\[ x^{-\frac{5}{3}} = \frac{1}{x^{\frac{5}{3}}} = \frac{1}{\sqrt[3]{x^5}} \][/tex]
Therefore, the expression equivalent to \( x^{-\frac{5}{3}} \) is:
[tex]\[ \frac{1}{\sqrt[3]{x^5}} \][/tex]
Among the given options, this corresponds to:
[tex]\[ \boxed{\frac{1}{\sqrt[3]{x^5}}} \][/tex]
