Problem 78 - Thomas Calculus

Question

Laws of Exponents a. Starting with the equation \(e^{x_{1}} e^{x_{2}}=e^{x_{1}+x_{2}}\) , derived in the text, show that \(e^{-x}=1 / e^{x}\) for any real number \(x\) . Then show that \(e^{x_{1} / e} e^{x_{2}}=e^{x_{1}-x_{2}}\) for any numbers \(x_{1}\) and \(x_{2}\) b. Show that \(\left(e^{x_{1}}\right)^{x_{2}}=e^{x_{1} x_{2}}=\left(e^{x_{2}}\right)^{x_{1}}\) for any numbers \(x_{1}\) and \(x_{2}\)

Step-by-Step Solution

Answer

a. \( e^{-x} = \frac{1}{e^x} \) and \( e^{x_1}/e^{x_2} = e^{x_1-x_2} \). b. \( \left(e^{x_1}\right)^{x_2} = e^{x_1 x_2} = \left(e^{x_2}\right)^{x_1} \).

Step 1: Understanding the Property

Given the property \( e^{x_{1}} e^{x_{2}} = e^{x_{1} + x_{2}} \), this means we add exponents when multiplying with the same base, \( e \).

Step 2: Apply the Property for Negative Exponent

We want to prove that \( e^{-x} = \frac{1}{e^{x}} \). Using the property, \( e^{x} e^{-x} = e^{x-x} = e^0 \). Since \( e^0 = 1 \), it follows that \( e^{-x} \) must equal \( \frac{1}{e^{x}} \) in order for \( e^{x}e^{-x} \) to be 1.

Step 3: Using the Property for Division of Exponentials

To prove \( e^{x_{1}} / e^{x_{2}} = e^{x_{1} - x_{2}} \), notice that dividing exponential terms with the same base involves subtracting the exponents: \( \frac{e^{x_{1}}}{e^{x_{2}}} = e^{x_{1}} imes e^{-x_{2}} = e^{x_{1} + (-x_{2})} = e^{x_{1} - x_{2}} \).

Step 4: Prove Equality of Powers and Products

We want to show \( \left(e^{x_{1}}\right)^{x_{2}} = e^{x_{1} x_{2}} \). By the power of a power property \((a^m)^n = a^{mn} \), thus \( \left(e^{x_{1}}\right)^{x_{2}} = e^{x_{1} x_{2}} \). Similarly, \(\left(e^{x_{2}}\right)^{x_{1}} = e^{x_{2} x_{1}} = e^{x_{1} x_{2}} \) as multiplication is commutative.

Key Concepts

Exponentiation

Exponentiation involves raising a number, known as the base, to the power of an exponent. In the expression \( a^n \), \( a \) is the base and \( n \) is the exponent, indicating how many times the base is used in a multiplication. For instance, \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \).
Exponentiation is fundamental in mathematics as it extends to various rules and properties used in simplifying expressions:
\( a^1 = a \): Any number raised to the power of 1 is the number itself.
\( a^0 = 1 \) (where \( a eq 0 \)): Any non-zero number raised to the power of 0 equals 1.
An important aspect is understanding recursive processes, where exponentiation can be multiplied, added, or subtracted for further calculations.
It provides the foundation for understanding more complex properties of exponents and their behaviors.

Properties of Exponents

The properties of exponents are essential rules that govern how exponential expressions are manipulated. Understanding these laws makes it easier to work with complex expressions and develop simplified forms.

• Product of Powers: For expressions with the same base, you can add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers: When dividing with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: To lift a power to another power, multiply the exponents: \( (a^m)^n = a^{mn} \).

These rules simplify complex algebraic expressions and equations. They help solve problems faster by breaking down exponential functions into manageable parts. The properties ensure that exponentiation's scope is vast and can be applied across all mathematical branches.

Negative Exponents

Negative exponents might initially seem confusing but are straightforward once broken down. They indicate reciprocals of the base raised to the opposite exponent.
For example, \( a^{-n} \) is interpreted as \( \frac{1}{a^n} \). This means the negative exponent moves the base to the denominator, flipping the power to positive. A simple illustration is \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \).
The logic rests on maintaining the equality \( a^m \times a^{-m} = 1 \), demonstrating that multiplication by a negative inverse balances to one. Negative exponents are incredibly useful in scientific notation and calculus, particularly in expressing very small quantities or inverse relationships. Embracing these exponent forms allows for greater flexibility and insight in mathematical problem-solving.