Sure, let's simplify the expression [tex]\((x^{-2}y^3z^8)^{-2}\)[/tex] step-by-step.
1. Start with the given expression:
[tex]\[(x^{-2}y^3z^8)^{-2}\][/tex]
2. Apply the exponent [tex]\(-2\)[/tex] to each factor inside the parentheses:
This step uses the power rule [tex]\((a^m)^n = a^{mn}\)[/tex].
[tex]\[ = (x^{-2})^{-2} \cdot (y^3)^{-2} \cdot (z^8)^{-2} \][/tex]
3. Simplify each factor:
- For [tex]\( (x^{-2})^{-2} \)[/tex], we have:
[tex]\[ (x^{-2})^{-2} = x^{(-2) \cdot (-2)} = x^4 \][/tex]
- For [tex]\( (y^3)^{-2} \)[/tex], we have:
[tex]\[ (y^3)^{-2} = y^{3 \cdot (-2)} = y^{-6} \][/tex]
- For [tex]\( (z^8)^{-2} \)[/tex], we have:
[tex]\[ (z^8)^{-2} = z^{8 \cdot (-2)} = z^{-16} \][/tex]
4. Combine these results:
[tex]\[ = x^4 \cdot y^{-6} \cdot z^{-16} \][/tex]
5. Rewrite the expression with positive exponents:
Since [tex]\( y^{-6} = \frac{1}{y^6} \)[/tex] and [tex]\( z^{-16} = \frac{1}{z^{16}} \)[/tex], we can express the product as a single fraction:
[tex]\[ = \frac{x^4}{y^6 \cdot z^{16}} \][/tex]
6. Final simplified form:
[tex]\[ = \frac{x^4}{y^6z^{16}} \][/tex]
Thus, the simplified form of [tex]\(\left(x^{-2} y^3 z^8\right)^{-2}\)[/tex] is:
[tex]\[
\boxed{\frac{x^4}{y^6 z^{16}}}
\][/tex]
