Solve the exponential equation: 9^(x-2) = 27^(-x+2)

Answer

To solve this exponential equation, we need to find a common base for both sides. 1. Identify common base: Both 9 and 27 are powers of 3. Specifically, 9 = 3^2 and 27 = 3^3. 2. Rewrite the equation: Substitute these into the original equation: (3^2)^(x-2) = (3^3)^(-x+2). 3. Apply the power of a power rule: Using the rule (a^m)^n = a^(m*n), we simplify the exponents: 3^(2*(x-2)) = 3^(3*(-x+2)). 4. Simplify the exponents: 3^(2x-4) = 3^(-3x+6). 5. Equate exponents: Since the bases are now the same, we can set the exponents equal to each other: 2x - 4 = -3x + 6. 6. Solve for x: Add 3x to both sides: 5x - 4 = 6. Add 4 to both sides: 5x = 10. Divide by 5: x = 2. The solution is x = 2.