To solve the equation \(3^{-2} \cdot 3^{-9} = 3^k\), we can use the property of exponents that states \(a^m \cdot a^n = a^{m+n}\).
So, we can combine the exponents on the left side:
\[ 3^{-2} \cdot 3^{-9} = 3^{-2 + (-9)} = 3^{-2 - 9} = 3^{-11} \]
Now we have:
\[ 3^{-11} = 3^k \]
Since the bases are the same, we can set the exponents equal to each other:
\[ k = -11 \]
Thus, the value of \(k\) is \(-11\).
The correct response is:
C -11