Final answer:
To solve the exponential equation 9ˣ⁺² = 27ˣ⁻¹, we simplify both sides of the equation using properties of exponents and equate the resulting exponents. The solution is x = 1.
Step-by-step explanation:
To solve the exponential equation 9ˣ⁺² = 27ˣ⁻¹, we need to first simplify both sides of the equation by using the properties of exponents. To do this, we recognize that 27 = 3³ and 9 = 3². So, our equation becomes (3²)ˣ⁺² = (3³)ˣ⁻¹. Next, we use the property of exponents where (a¹)ᵇ = aᵇ, to rewrite the equation as 3²ˣ × 3² = 3³ˣ × 3⁻¹. Now we can equate the exponents on both sides of the equation: 2x + 2 = 3x - 1. Solving for x, we subtract 2x from both sides to get -1 = x - 2.
Adding 2 to both sides gives us x = 1. So, the exact solution to the exponential equation is x = 1.