Problem 46 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $$e^{x y}=e^{x} e^{y}$$

Step-by-Step Solution

Answer

The statement \(e^{xy} = e^x e^y\) is true, as it follows the first property of exponentiation. This property states that if two exponents are added together with the same base, it is equivalent to multiplying the exponential functions with the same base and the separate exponents individually. Since "e" is the common base in this statement, the given expression holds true.

Step 1: Recall properties of exponentiation

The properties of exponentiation include: 1. \(a^{m+n} = a^m a^n\) 2. \(a^{-m} = \frac{1}{a^m}\) 3. \((a^m)^{n} = a^{m \cdot n}\) It is important to remember that these properties hold true for all positive real numbers a, and any real numbers m and n.

Step 2: Apply the properties to our statement

Let's replace \(a\) with \(e\) in the first property of exponentiation: 1. \(e^{m+n} = e^m e^n\) Now let's compare this property with the given statement \(e^{xy} = e^x e^y\): - \(m + n = xy\)

Step 3: Analyze the relationship between the two expressions

The statement appears to follow the properties of exponentiation, but it is important to understand the context. The given statement corresponds to the first property of exponentiation with "m" being replaced by "x" and "n" being replaced by "y". In this case, it appears the statement is true.

Step 4: Provide an explanation

The statement \(e^{xy} = e^x e^y\) is true because it follows the first property of exponentiation. Exponential functions with a common base can be rewritten as a product of two separate exponential functions, each with an individual exponent added together. In this case, the common base "e" allows for the statement to be true.

Key Concepts

Property of Exponents

Understanding the properties of exponents is essential for mastering exponential expressions. One of the key rules is that for any positive real number \(a\) and real numbers \(m\) and \(n\), the expression \(a^{m+n} = a^m a^n\) holds. This means you can add the exponents when you multiply like bases. This property underpins many exponential calculations and simplifies complex exponential expressions into manageable parts.

• When you see a shared base, like \(e\) in \(e^{m+n}\), you know it can be rewritten as \(e^m \times e^n\).
• Common scenarios involve this property, especially in algebra and calculus.

Recognizing and applying this correctly helps solve various equations and problems involving exponents.

Mathematical Proof

Mathematical proof provides the foundation for verifying statements like \(e^{xy} = e^x e^y\). A proof uses logic and existing rules to demonstrate the truth or falsehood of a given statement. In this exercise, understanding why \(e^{xy}\) is not equivalent to \(e^x e^y\) requires a logical examination of exponent rules.

• This involves comparing the structure of the expressions with known properties: \(e^{m+n} = e^m e^n\) but not \(e^{xy}\).
• Proof showcases that although bases are the same, the multiplication inside the exponent differs from addition, deviating from exponent properties.

This logical approach ensures accuracy and precision in mathematics. Proof is essential to establish the reliability of a mathematical statement.

Exponential Functions

Exponential functions, such as \(e^x\), are vital in mathematics, appearing in growth models, calculus, and scientific equations. An exponential function has the form \(a^x\) where \(a\) is a positive constant. The base "\(e\)", roughly equal to 2.718, is especially significant in natural processes and mathematical applications.

• These functions model scenarios where change is proportional to the function's current value, like population growth or radioactive decay.
• Exponential functions are unique because their rate of growth is rapid and consistent, different from linear or polynomial functions.

grasping their behavior and principles helps in analyzing broader mathematical and real-world phenomena. They are key to understanding trends and changes in complex systems.