To determine which of the choices correctly represents the expression [tex]\(\left(\sqrt[6]{x^7 z^{-2}}\right)^5\)[/tex] with a rational exponent, we will start by simplifying the expression step-by-step.
1. Start with the given expression:
[tex]\[
\left(\sqrt[6]{x^7 z^{-2}}\right)^5
\][/tex]
2. Rewrite the nested radical using rational exponents. The sixth root of any number or expression [tex]\(a\)[/tex] can be written as [tex]\(a^{\frac{1}{6}}\)[/tex]:
[tex]\[
\sqrt[6]{x^7 z^{-2}} = (x^7 z^{-2})^{\frac{1}{6}}
\][/tex]
3. Raising a power to another power, we multiply the exponents:
[tex]\[
\left((x^7 z^{-2})^{\frac{1}{6}}\right)^5 = (x^7 z^{-2})^{\frac{1}{6} \cdot 5} = (x^7 z^{-2})^{\frac{5}{6}}
\][/tex]
4. Apply the exponent to both factors inside the parentheses:
[tex]\[
\left(\frac{x^7}{z^2}\right)^{\frac{5}{6}}
\][/tex]
This matches with the answer choice (A). Hence, the correct answer is:
[tex]\[
\left(\frac{x^7}{z^2}\right)^{\frac{5}{6}}
\][/tex]
