Certainly! Let's break down the given problem step by step to determine the equivalent expression for [tex]\(\left(a^{-7} \cdot b^{-2}\right)^{-9}\)[/tex].
1. Start with the given expression:
[tex]\[
\left(a^{-7} \cdot b^{-2}\right)^{-9}
\][/tex]
2. Use the property of exponents [tex]\((x^m)^n = x^{m \cdot n}\)[/tex] separately for each term within the parentheses.
3. Simplify the exponent for [tex]\(a^{-7}\)[/tex]:
[tex]\[
(a^{-7})^{-9} = a^{(-7) \cdot (-9)} = a^{63}
\][/tex]
Explanation: When you raise a power to another power, you multiply the exponents. Here, [tex]\(-7\)[/tex] multiplied by [tex]\(-9\)[/tex] gives [tex]\(63\)[/tex].
4. Simplify the exponent for [tex]\(b^{-2}\)[/tex]:
[tex]\[
(b^{-2})^{-9} = b^{(-2) \cdot (-9)} = b^{18}
\][/tex]
Explanation: Similarly, [tex]\(-2\)[/tex] multiplied by [tex]\(-9\)[/tex] gives [tex]\(18\)[/tex].
5. Combine the simplified terms:
[tex]\[
\left(a^{-7} \cdot b^{-2}\right)^{-9} = a^{63} \cdot b^{18}
\][/tex]
6. Identify the equivalent expression from the given choices:
- (A) [tex]\(\frac{b^{18}}{a^{63}}\)[/tex]
- (B) [tex]\(\frac{a^{63}}{b^{18}}\)[/tex]
- (C) [tex]\(a^{63} \cdot b^{18}\)[/tex]
The correct answer is:
[tex]\[
\boxed{a^{63} \cdot b^{18}}
\][/tex]
So the equivalent expression for [tex]\(\left(a^{-7} \cdot b^{-2}\right)^{-9}\)[/tex] is [tex]\(a^{63} \cdot b^{18}\)[/tex]. This matches option (c).
