To determine the correct representation of the expression [tex]\(\left(\sqrt[6]{x^7 z^{-2}}\right)^5\)[/tex] with a rational exponent, we will follow the steps below:
1. Simplify the Inside of the Root Expression:
The given expression inside the root is [tex]\(x^7 z^{-2}\)[/tex].
2. Express the Root Using Rational Exponents:
Taking the sixth root of [tex]\(x^7 z^{-2}\)[/tex] can be rewritten using rational exponents:
[tex]\[
\sqrt[6]{x^7 z^{-2}} = (x^7 z^{-2})^{1/6}
\][/tex]
3. Raise the Expression to the Fifth Power:
Raise the simplified form to the fifth power:
[tex]\[
\left( (x^7 z^{-2})^{1/6} \right)^5
\][/tex]
4. Apply the Exponent Rules:
When raising a power to another power, multiply the exponents:
[tex]\[
(x^7 z^{-2})^{(1/6) \cdot 5}
\][/tex]
Simplify the multiplication of the exponents:
[tex]\[
(x^7 z^{-2})^{5/6}
\][/tex]
5. Simplify the Expression:
Combine the exponents inside the base using the property [tex]\(a^{m/n} = a^{(m/n)}\)[/tex]:
[tex]\[
\left(\frac{x^7}{z^2}\right)^{5/6}
\][/tex]
So, after these transformations, we simplify the original expression [tex]\(\left(\sqrt[6]{x^7 z^{-2}}\right)^5\)[/tex] to:
[tex]\[
\left(\frac{x^7}{z^2}\right)^{5/6}
\][/tex]
Analyzing the given options:
- A. [tex]\(\left(\frac{x^7}{2^2}\right)^{\frac{8}{6}}\)[/tex]
- B. [tex]\(\left(\frac{x^7}{x^2}\right)^{\frac{5}{5}}\)[/tex]
- C. [tex]\(\left(\frac{x^7 z}{2}\right)^{\frac{5}{6}}\)[/tex]
- D. [tex]\(\left(\frac{x^7 z}{2}\right)^{\frac{6}{5}}\)[/tex]
None of these expressions exactly match our result. The closest to the exponents and the form we computed [tex]\( \left(\frac{x^7}{z^2}\right)^{5/6} \)[/tex] is also not present in the list. This confirms the expected simplified form as analyzed but does not correspond directly to any provided options.
Therefore, given the detailed rationalization and simplification detailed above, none of the given options A, B, C, or D strictly represent the expression correctly with their rational exponents.
