Sure, let's solve the equation step-by-step.
We start with the given equation:
[tex]\[ 3^{-1} \cdot 3^{-6} = 3^2 \][/tex]
First, we use the property of exponents that states:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
Applying this to the left side of the equation, we get:
[tex]\[ 3^{-1} \cdot 3^{-6} = 3^{-1 + (-6)} = 3^{-7} \][/tex]
So, our equation now looks like this:
[tex]\[ 3^{-7} = 3^2 \][/tex]
For the two sides of the equation to be equal, the exponents must also be equal. Therefore, we have:
[tex]\[ -7 = 2 \][/tex]
This shows that the missing exponent that completes the equation is indeed:
[tex]\[ -7 \][/tex]
So, the correct answer to complete the equation is:
[tex]\[ -7 \][/tex]
