Solve the exponential equation by expressing each side as a power of the same base and then equating exponents. \[ 9^{\frac{x-9}{4}}=\sqrt{9} \] The solution set is \( \{\square\} \)

To solve the equation \( 9^{\frac{x-9}{4}}=\sqrt{9} \), we start by rewriting both sides with the same base. We know that \( 9 \) can be expressed as \( 3^2 \). Thus, we have: \[ 9^{\frac{x-9}{4}} = (3^2)^{\frac{x-9}{4}} = 3^{\frac{2(x-9)}{4}} = 3^{\frac{x-9}{2}} \] Next, we rewrite \( \sqrt{9} \) as \( 9^{1/2} \) or \( (3^2)^{1/2} = 3^{2 \cdot \frac{1}{2}} = 3^1 \). Now we set the exponents equal, since the bases are the same: \[ \frac{x-9}{2} = 1 \] To solve for \( x \), we multiply both sides by 2: \[ x - 9 = 2 \] Adding 9 to both sides, we find: \[ x = 11 \] Thus, the solution set is \( \{11\} \).