Question

In Exercises 3–12, simplify the expression. $$ \left(4 e^{-2 x}\right)^3 $$

Short answer

The simplified expression for \((4 e^{-2x})^3\) is \(64e^{-6x}\).

Step-by-step solution

Breaking down the expression

First, break down the expression to separate the base number and the exponent. The expression \((4 e^{-2x})^3\) becomes \(4^3 * (e^{-2x})^3\).

Applying the power rule and exponential rule

Apply the power rule to \(4^3\) and the exponent rule \(a^(m*n) = a^(m)*a^(n)\) to \((e^{-2x})^3\). This leads to \(64 * e^{-6x}\).

Combining results

The final step is to combine the results from step 2 to get the final simplified expression, which is \(64e^{-6x}\).

Key concepts

Power Rule

The power rule is a fundamental concept in algebra that deals with raising an expression to an exponent. It states that when you have a power raised to another power, you multiply the exponents. In mathematical terms, for any real number 'a' and integers 'm' and 'n', the rule is expressed as \( (a^m)^n = a^{m \times n} \).

For instance, in the given exercise \( (4 e^{-2x})^3 \), we see this rule applied to \( 4^3 \), which simplifies to \( 4 \times 4 \times 4 = 64 \). It's important to understand and apply this rule correctly as it not only helps to simplify expressions but also lays the groundwork for more advanced algebraic operations.

Exponent Rules

Exponent rules, also known as laws of exponents, are used to simplify expressions involving powers. These rules are essential when working with exponential expressions. There are several exponent rules, but the most relevant for simplifying the given expression is the rule for multiplying powers with the same base: \( a^m \times a^n = a^{m + n} \). The power rule mentioned earlier is a specific case of this general exponent rule.

In the context of our exercise, we applied an exponent rule to \( e^{-2x} \), raised to the power of three. We simplify \( (e^{-2x})^3 \), using the rule that states \( a^{m\times n} = (a^{m})^n \), which simplifies to \( e^{-6x} \). Knowing these rules helps students manage more complex expressions and perform algebraic simplification effectively.

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent. Formally, an exponential function can be written as \( f(x) = a^{x} \), where 'a' is a constant, and 'a' is greater than 0. In our given expression \( 4e^{-2x} \), 'e' represents the base of the natural logarithm, approximately equal to 2.71828, and is a commonly used base in exponential functions.

Understanding exponential functions is crucial in various fields such as finance, physics, and computer science. They are used to model growth and decay processes, like population growth or radioactive decay. Simplifying exponential functions, like we did in the exercise, is an important skill that empowers students to handle real-world problems involving exponentially changing quantities.

Algebraic Simplification

Algebraic simplification is the process of making a complex algebraic expression easier to understand and work with. This involves using the exponent rules, power rule, and other algebraic properties to condense and combine like terms. The goal is to write the expression in its simplest form without changing its value.

In the original example, we perform algebraic simplification by applying the power and exponent rules to achieve the final simplified result of \( 64e^{-6x} \). This process not only makes the expression cleaner and more straightforward but also prepares it for further mathematical operations such as differentiation, integration, or graphing. Mastery of algebraic simplification is an invaluable tool in a student's math toolkit, ensuring clarity and precision in problem-solving.