To solve the expression [tex]\(\left(7^{-2} \cdot 7^3\right)^{-3}\)[/tex], we need to follow the laws of exponents step-by-step.
1. Simplify the expression inside the parentheses:
[tex]\[
7^{-2} \cdot 7^3
\][/tex]
When multiplying terms with the same base, we add the exponents:
[tex]\[
7^{-2 + 3} = 7^1 = 7
\][/tex]
2. Raise the simplified result to the power of -3:
[tex]\[
(7^1)^{-3}
\][/tex]
When raising a power to another power, we multiply the exponents:
[tex]\[
7^{1 \cdot -3} = 7^{-3}
\][/tex]
3. Rewrite the expression with a positive exponent:
[tex]\[
7^{-3} = \frac{1}{7^3}
\][/tex]
Comparing this to the given choices:
- [tex]\( -76 \)[/tex]
- [tex]\(\frac{1}{7^3}\)[/tex]
- [tex]\(-\frac{1}{7^6}\)[/tex]
- [tex]\(7^{18}\)[/tex]
The correct equivalent expression is [tex]\(\frac{1}{7^3}\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{\frac{1}{7^3}}
\][/tex]
