To simplify the given exponential expression [tex]\((x^{-2})^9\)[/tex], let's follow a step-by-step process:
1. Understand the expression:
[tex]\[
(x^{-2})^9
\][/tex]
2. Apply the power of a power property of exponents:
When you raise a power to another power, you multiply the exponents. The property is:
[tex]\[
(a^m)^n = a^{m \cdot n}
\][/tex]
3. Apply this property to our expression:
[tex]\[
(x^{-2})^9 = x^{-2 \cdot 9} = x^{-18}
\][/tex]
4. Convert the expression to use a positive exponent:
To express [tex]\(x^{-18}\)[/tex] with a positive exponent, you use the property:
[tex]\[
a^{-m} = \frac{1}{a^m}
\][/tex]
Therefore:
[tex]\[
x^{-18} = \frac{1}{x^{18}}
\][/tex]
So the simplified form of the given exponential expression [tex]\((x^{-2})^9\)[/tex] is:
[tex]\[
\boxed{x^{-18}}
\][/tex]
And, converted to exponential notation with a positive exponent, it remains:
[tex]\[
\boxed{x^{-18}}
\][/tex]
