Sure, let's solve the equation step-by-step.
The given equation is:
[tex]\[ 3^3 \cdot 3^{-6} = 3^2 \][/tex]
To solve this, notice that the left-hand side combines the exponents of the same base (3). According to the properties of exponents, when you multiply powers of the same base, you add the exponents together:
[tex]\[ 3^3 \cdot 3^{-6} = 3^{3 + (-6)} \][/tex]
So, the combined exponent on the left-hand side is:
[tex]\[ 3 + (-6) = -3 \][/tex]
Therefore, the equation becomes:
[tex]\[ 3^{-3} = 3^2 \][/tex]
For the equation to hold true, we need to match the exponents. To do that, we equate the combined exponent to the exponent on the right-hand side, which is 2. So:
[tex]\[ -3 + \text{(missing exponent)} = 2 \][/tex]
To find the missing exponent, we solve for it:
[tex]\[ \text{(missing exponent)} = 2 + 3 \][/tex]
[tex]\[ \text{(missing exponent)} = 5 \][/tex]
So the completed equation is:
[tex]\[ 3^3 \cdot 3^{-6} \cdot 3^5 = 3^2 \][/tex]
The missing exponent that completes the equation is:
5
Therefore, the answer is 5.
