In Exercises 81–100, evaluate or simplify each expression without using a calculator. In (1/e⁶)

Verified step by step guidance

1. Recognize that the expression is \( \frac{1}{e^6} \), which is a fraction with the base \( e \) raised to the power 6 in the denominator.
2. Recall the property of exponents that \( \frac{1}{a^n} = a^{-n} \), where \( a \) is a nonzero number and \( n \) is an integer.
3. Apply this property to rewrite \( \frac{1}{e^6} \) as \( e^{-6} \).
4. Understand that \( e^{-6} \) is the simplified form of the original expression, representing the exponential function with a negative exponent.
5. No further simplification is needed unless the problem asks for a decimal approximation, which is not allowed here.

Key Concepts

Properties of Exponents

Understanding the rules of exponents is essential, such as how to handle negative exponents and powers of e. For example, e^{-6} is equivalent to 1/e^6, which helps in rewriting and simplifying expressions involving exponential terms.

Simplifying Exponential Expressions

Simplifying expressions like (1/e^6) involves recognizing that it can be rewritten using negative exponents as e^{-6}. This skill allows for easier manipulation and combination of exponential terms without a calculator.

Natural Exponential Function (e)

The constant e is the base of natural logarithms and appears frequently in algebra and calculus. Knowing its properties, such as e^0 = 1 and how to work with its powers, is crucial for simplifying expressions involving e.