To simplify the expression [tex]\(\left(z^7 z^{-5}\right)^{-2}\)[/tex], follow these steps:
1. Combine like terms within the parenthesis:
- When you have [tex]\(z^7\)[/tex] and [tex]\(z^{-5}\)[/tex] together, you add the exponents: [tex]\(z^7 z^{-5}\)[/tex].
[tex]\[
z^7 \cdot z^{-5} = z^{7-5} = z^2
\][/tex]
2. Apply the exponent outside the parenthesis:
- Now that we have simplified the expression inside the parenthesis to [tex]\(z^2\)[/tex], we need to deal with the external exponent [tex]\(-2\)[/tex]:
[tex]\[
(z^2)^{-2}
\][/tex]
3. Multiply the exponents:
- To handle the exponent [tex]\(-2\)[/tex] applied to [tex]\(z^2\)[/tex], multiply the exponents:
[tex]\[
(z^2)^{-2} = z^{2 \cdot (-2)} = z^{-4}
\][/tex]
4. Rewrite using positive exponents (if necessary):
- An exponent of [tex]\(-4\)[/tex] can be written as a reciprocal:
[tex]\[
z^{-4} = \frac{1}{z^4}
\][/tex]
Based on these steps, the simplified form of the expression [tex]\(\left(z^7 z^{-5}\right)^{-2}\)[/tex] is [tex]\(\frac{1}{z^4}\)[/tex]. Thus, the correct answer is:
[tex]\(\frac{1}{z^4}\)[/tex].
