Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4.e^2x - 6e^x + 8 = 0

Exponential equations involve variables in the exponent, such as e^x. To solve these equations, one often uses substitution to simplify the expression. For example, if we let y = e^x, the equation can be transformed into a quadratic form, making it easier to solve.

The quadratic formula is a method for solving quadratic equations of the form ax^2 + bx + c = 0. It states that the solutions for x can be found using x = (-b ± √(b² - 4ac)) / (2a). This formula is essential for finding exact solutions, especially when the equation does not factor easily.

Irrational solutions are those that cannot be expressed as simple fractions and often involve square roots or other non-repeating decimals. In contrast, exact solutions are expressed in their simplest radical form. Understanding how to convert between these forms is crucial for providing answers in the required format.