Certainly! Let's solve this step-by-step.
We are given the equation:
[tex]\[3^1 \cdot 3^{-6} \cdot 3^x = 3^2\][/tex]
1. We start by combining the exponents of the left-hand side since they have the same base.
[tex]\[3^{1 + (-6) + x} = 3^2\][/tex]
2. Simplify the exponents inside the parentheses:
[tex]\[3^{1 - 6 + x} = 3^2\][/tex]
3. Combine the constants in the exponent:
[tex]\[3^{-5 + x} = 3^2\][/tex]
4. Since the bases are the same, we can set the exponents equal to each other:
[tex]\[-5 + x = 2\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[x = 2 + 5\][/tex]
[tex]\[x = 7\][/tex]
The missing exponent is:
[tex]\[x = 7\][/tex]
Thus, 7 is the value that completes the equation.
