Problem 78 - University Calculus: Early Transcendentals

Question

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

The expressions are equal by properties of exponents: \( (e^{x_1})^{x_2} = e^{x_1 x_2} = (e^{x_2})^{x_1} \).

Step 1: Understanding the Problem

We need to show that \( (e^{x_1})^{x_2} = e^{x_1 x_2} = (e^{x_2})^{x_1} \) for any numbers \( x_1 \) and \( x_2 \). This involves comparing expressions by applying properties of exponents.

Step 2: Applying Exponent Rules

Recall that for any base \( a \) and exponents \( m \) and \( n \), we have the rule \( (a^m)^n = a^{m \cdot n} \). This means \( (e^{x_1})^{x_2} = e^{x_1 \cdot x_2} \) and \( (e^{x_2})^{x_1} = e^{x_2 \cdot x_1} \). Since multiplication is commutative \( x_1 \cdot x_2 = x_2 \cdot x_1 \), we have both equal to \( e^{x_1 x_2} \).

Step 3: Verification of Both Sides

We need to confirm that both expressions \( (e^{x_1})^{x_2} \) and \( (e^{x_2})^{x_1} \) simplify to \( e^{x_1 x_2} \). From the previous step, \( (e^{x_1})^{x_2} = e^{x_1 x_2} \) and \( (e^{x_2})^{x_1} = e^{x_2 x_1} = e^{x_1 x_2} \). This shows that the expressions are indeed equal.

Step 4: Conclusion

Since both expressions simplify to \( e^{x_1 x_2} \), we have shown that \( (e^{x_1})^{x_2} = e^{x_1 x_2} = (e^{x_2})^{x_1} \) for any real numbers \( x_1 \) and \( x_2 \).

Key Concepts

Exponent Rules

Exponent rules are fundamental in simplifying and solving expressions involving powers. One of the basic exponent rules is the power of a power rule. This rule states that when you raise an exponential expression to another exponent, you multiply the exponents together. For example, for a base \(a\) and exponents \(m\) and \(n\), the rule is expressed as

• \((a^m)^n = a^{m \cdot n}\)

This rule is particularly handy when dealing with nested exponents, like in the problem \((e^{x_1})^{x_2}\). Using the power of a power rule, we can simplify this expression to

• \(e^{x_1 \cdot x_2}\)

This simplifies computation and shows how exponents can be manipulated through multiplication of their indices. Remember that exponent rules help maintain the integrity of equations while transforming expressions into simpler and more manageable forms.

Commutative Property of Multiplication

The commutative property of multiplication is a vital property in arithmetic that simplifies calculations and proofs. It states that changing the order of factors does not change the product's value. Mathematically, this property is expressed as

• \(a \cdot b = b \cdot a\)

for any numbers \(a\) and \(b\). Applying this property in the context of exponent rules, especially in our original exercise, helps to demonstrate equality between two expressions \((e^{x_1})^{x_2}\) and \((e^{x_2})^{x_1}\).
By recognizing that

• \(x_1 \cdot x_2 = x_2 \cdot x_1\)

we can affirm that

• \(e^{x_1 x_2} = e^{x_2 x_1}\)

using both the power of a power rule and the commutative property, confirming that the expressions are indeed identical. The commutative property underlines the flexibility in handling multiplication, especially when linked with exponentiation.

Exponential Functions

Exponential functions are fundamental mathematical expressions that incorporate exponents and are widely used in various scientific fields, including biology, economics, and physics. An exponential function is commonly written as

• \(f(x) = a \, \cdot \, b^x\)

where \(a\) is a constant, and \(b\) is the base raised to the variable \(x\). These functions are unique because their rate of change is proportional to their current value, making them suitable for modeling growth and decay processes. In the context of solving exponential equations, like those seen in the original exercise involving \(e\), the natural exponential base of \(e\) is frequently used.
It is crucial for expressing natural phenomena and processes that expand continuously.

• The number \(e\) is approximately 2.71828...

Examples include population growth, radioactive decay, and compounded interest. Understanding exponential functions and their properties allows us to solve complex real-world problems and establish relationships between variables through mathematical modeling.