Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Simplify the expression. \(\frac{\left(e^{x}-e^{-x}\right)^{2}-\left(e^{x}+e^{-x}\right)^{2}}{\left(e^{x}+e^{-x}\right)^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-4}{\left(e^{x}+e^{-x}\right)^{2}} \).

Step 1: Identify the Expression

The given expression is \( \frac{\left(e^{x}-e^{-x}\right)^{2}-\left(e^{x}+e^{-x}\right)^{2}}{\left(e^{x}+e^{-x}\right)^{2}} \). You need to simplify this by dealing with the squared terms and performing algebraic operations.

Step 2: Expand the Squared Terms

Expand \( \left(e^{x}-e^{-x}\right)^{2} \) as \( e^{2x} - 2 + e^{-2x} \) and \( \left(e^{x}+e^{-x}\right)^{2} \) as \( e^{2x} + 2 + e^{-2x} \).

Step 3: Subtract the Expanded Results

Subtract the second expansion from the first: \( (e^{2x} - 2 + e^{-2x}) - (e^{2x} + 2 + e^{-2x}) = -4 \). This is because \( e^{2x} - e^{2x} = 0\), \( e^{-2x} - e^{-2x} = 0 \) and \(-2 - 2 = -4\).

Step 4: Simplify the Expression Further

Now substitute back into the original expression: \( \frac{-4}{\left(e^{x}+e^{-x}\right)^{2}} \). The expression is now simplified to this form.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In most cases, the base is denoted as \( e \), known as Euler's number, which is approximately equal to 2.71828. Exponential functions have numerous practical applications such as in growth and decay problems, and they often appear in calculus and differential equations.

One peculiarity of exponential functions is that they grow very rapidly. For example, \( e^x \) increases much faster than a linear function as \( x \) becomes large. Conversely, \( e^{-x} \) decreases and approaches zero as \( x \) becomes large.

When dealing with exponential expressions like \( e^x \) and \( e^{-x} \), it is useful to remember properties such as \( (e^x)^n = e^{nx} \) and \( e^x \cdot e^{-x} = 1 \). These properties come in handy during simplification processes, as seen in the given expression.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators like addition, subtraction, multiplication, and division. The primary goal with algebraic expressions is often to simplify them to their simplest form.

In the case of the given expression, we observe inner expressions \((e^x - e^{-x})\) and \((e^x + e^{-x})\), both squared. It's important to expand these squared terms correctly using the formula \((a+b)^2 = a^2 + 2ab + b^2\). Likewise, \((a-b)^2 = a^2 - 2ab + b^2\).

Knowing how to deal with these expansions can significantly simplify algebraic work. This exercise specifically requires careful attention to the signs of each term when expanding and simplifying.

Simplification Steps

Simplifying algebraic expressions involves transforming a complex expression into a simpler one without changing its value. Let's look at the steps involved in this simplification:

• Identify the Expression: Recognize the need for simplification and identify components that need to be simplified, as done in Step 1 of the solution.
• Expand the Squared Terms: Use algebraic identities to expand \((e^x - e^{-x})^2\) and \((e^x + e^{-x})^2\). This is crucial for revealing the underlying structure of the expression.
• Subtract the Expanded Results: Perform algebraic operations to combine like terms. Reducing terms to their simplest forms can often cancel out components, as seen when subtracting the expanded terms.
• Simplify Further: Once you have a simplified numerator, incorporate it back into the original expression. This final step often results in a much simpler ratio to handle, such as \( \frac{-4}{(e^x + e^{-x})^2} \).

These steps are typical of simplifying complex expressions and are essential skills in algebra to make expressions more manageable and understandable.