In Exercises 81–100, evaluate or simplify each expression without using a calculator. In e^9x

Verified step by step guidance

1. Identify the expression given: \( e^{9x} \). This is an exponential expression where the base is \( e \), the natural exponential constant, and the exponent is \( 9x \).
2. Recall the property of exponents that states \( (a^m)^n = a^{m \cdot n} \) and \( a^{m+n} = a^m \cdot a^n \). These properties help in simplifying exponential expressions.
3. Since the expression is \( e^{9x} \), consider if it can be rewritten using the property \( e^{a+b} = e^a \cdot e^b \). For example, if \( 9x \) can be expressed as a sum, you can split the exponent accordingly.
4. If the problem requires evaluating or simplifying without a calculator, check if \( x \) has a given value or if the expression can be factored or rewritten in a simpler form using exponent rules.
5. If no further information about \( x \) is given, the expression \( e^{9x} \) is already in its simplest form, and no further simplification is possible without additional context.

Key Concepts

Exponential Functions

Exponential functions involve expressions where a constant base is raised to a variable exponent, such as e^(9x). The base e is a special irrational number approximately equal to 2.718, commonly used in natural growth and decay models.

Properties of Exponents

Understanding how to manipulate exponents is essential, including rules like e^(a+b) = e^a * e^b and (e^a)^b = e^(ab). These properties allow simplification and evaluation of expressions without a calculator.

Simplification Techniques

Simplifying expressions involves rewriting them in a more manageable form by applying algebraic rules and exponent properties. This skill helps in evaluating expressions like e^(9x) by breaking down or combining terms logically.