Problem 39 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 38 through 44 find all \(x\) for which each equation is true. $$ e^{x^{3}}=\left(e^{x}\right)^{3} $$

Step-by-Step Solution

Answer

The solutions to the equation \(e^{x^3} = (e^x)^3\) are \(x = 0\), \(x = \sqrt{3}\), and \(x = -\sqrt{3}\).

Step 1: Simplify the equation

The equation given is \(e^{x^3} = (e^x)^3\). Rewrite \((e^x)^3\) as \(e^{3x}\) to make the equation easier to work with. So, the equation becomes \(e^{x^3} = e^{3x}\). This simplification is possible due to the property of exponentials that states \(a^{mn} = (a^m)^n\).

Step 2: Equate the exponents

In this expression, both sides have the same base \(e\). So the exponents must be equal. Setting the exponents equal to each other gives \(x^3 = 3x\).

Step 3: Rearrange the equation

Rearrange the equation to prepare it for solving. Moving all terms to one side, we have \(x^3 - 3x = 0\).

Step 4: Factor the expression

The expression can be factored into \(x(x - \sqrt{3})(x + \sqrt{3}) = 0\). This allows to set each factor equal to zero and solve for \(x\).

Step 5: Solve for \(x\)

Setting each factor equal to zero gives the following equations: \(x = 0\), \(x = \sqrt{3}\), and \(x = -\sqrt{3}\). These are the values of \(x\) that solve the original equation.

Key Concepts

Equation Solving

Equation solving is an essential skill in mathematics, helping us find the unknown values that make an equation true. In our problem, we're tasked with solving the equation \( e^{x^3} = (e^x)^3 \). To solve this, we utilize properties of exponents, which allow us to rewrite this equation. By identifying that both sides have the base \( e \), we can focus on the exponents themselves.

• Step 1 is noticing the exponent rule: \( (a^m)^n = a^{mn} \). This lets us write \((e^x)^3\) as \(e^{3x}\).
• Next, since \( e^{x^3} = e^{3x} \) has the same base, set the exponents equal: \( x^3 = 3x \).

This method reduces a tricky exponential equation to something easier - a polynomial equation, opening the door to more straightforward solving techniques.

Factoring Expressions

Factoring expressions is a key algebraic tool to simplify equations and solve them more easily. In our equation, \( x^3 = 3x \), we can rearrange it to \( x^3 - 3x = 0 \). This already hints at a possible solution: factoring.

• Factor out the common \( x \): \( x(x^2 - 3) = 0 \).
• This can be further factored into \( x(x - \sqrt{3})(x + \sqrt{3}) = 0 \).

Through factoring, the original polynomial breaks down into linear components. Each factor corresponds to a potential solution for \( x \). By setting each factor equal to zero, you find the values of \( x \) that solve the equation.

Properties of Exponents

Understanding the properties of exponents is crucial when dealing with exponential functions. These properties simplify complex equations, making them manageable. Primarily, they involve rules for multiplying and dividing exponents, and how they apply to powers of powers, like in our problem:

• The key property here is \( (a^m)^n = a^{mn} \), allowing us to rewrite \((e^x)^3\) as \(e^{3x}\).
• Another important property used implicitly is if \(a^b = a^c\), then \(b = c\), given the bases are the same.

By applying these properties, the structure of the equation changes, and exponential equations can be transformed back into algebraic ones, an essential technique in both algebra and calculus.