Select the correct answer.

Which of the following correctly represents the expression [tex]\left(\sqrt[6]{x^7 z^{-2}}\right)^5[/tex] with a rational exponent?

A. [tex]\left(\frac{x^7}{z^2}\right)^{\frac{5}{6}}[/tex]

B. [tex]\left(\frac{x^7}{z^2}\right)^{\frac{6}{6}}[/tex]

C. [tex]\left(\frac{x^7 z}{2}\right)^{\frac{3}{6}}[/tex]

D. [tex]\left(\frac{x^7 z}{2}\right)^{\frac{6}{5}}[/tex]



Answer :

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]