Question

Simplify as far as possible: (a) \(\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{2}+\frac{\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}\) (b) \(\mathrm{e}^{x}+\frac{1}{\mathrm{e}^{x}}-\mathrm{e}^{-x}\) (c) \(\frac{\mathrm{e}^{2 x}+\mathrm{e}^{x}}{\mathrm{e}^{x}}-1\) (d) \(\mathrm{e}^{3 x}\left(\mathrm{e}^{-2 x}-\mathrm{e}^{-3 \mathrm{x}}\right)+1\)

Short answer

Question: Simplify the following expressions and identify if any two expressions are equal: (a) \(\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{2}+\frac{\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}\) (b) \(\mathrm{e}^{x}+\frac{1}{\mathrm{e}^{x}}-\mathrm{e}^{-x}\) (c) \(\frac{\mathrm{e}^{2 x}+\mathrm{e}^{x}}{\mathrm{e}^{x}}-1\) (d) \(\mathrm{e}^{3 x}\left(\mathrm{e}^{-2 x}-\mathrm{e}^{-3 x}\right)+1\) Answer: All four expressions simplify to \(\mathrm{e}^{x}\). Therefore, they are all equal to each other.

Step-by-step solution

Combine the fractions

We can add the fractions by adding the numerators and keeping the common denominator. Result: \(\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}+\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}\)

Simplify the numerator

The terms \(\mathrm{e}^{x}\) and \(\mathrm{e}^{-x}\) will cancel out. Result: \(\frac{2\mathrm{e}^{x}}{2}\)

Cancel the common factors

The factors of 2 will cancel out, and we are left with just \(\mathrm{e}^{x}\). Result: \(\mathrm{e}^{x}\) (b) Simplify \(\mathrm{e}^{x}+\frac{1}{\mathrm{e}^{x}}-\mathrm{e}^{-x}\)

Express terms with the same exponent

Rewrite the second term with the same exponent as the third term: \(\frac{1}{\mathrm{e}^{x}} = \mathrm{e}^{-x}\) Result: \(\mathrm{e}^{x}+\mathrm{e}^{-x}-\mathrm{e}^{-x}\)

Simplify the expression

The terms \(\mathrm{e}^{-x}\) will cancel out, and we are left with just \(\mathrm{e}^{x}\). Result: \(\mathrm{e}^{x}\) (c) Simplify \(\frac{\mathrm{e}^{2 x}+\mathrm{e}^{x}}{\mathrm{e}^{x}}-1\)

Factor out common exponent in numerator

Factor out \(\mathrm{e}^{x}\) in the numerator. Result: \(\frac{\mathrm{e}^{x}(\mathrm{e}^{x}+1)}{\mathrm{e}^{x}}-1\)

Cancel the common factors

The factors of \(\mathrm{e}^{x}\) will cancel out. Result: \(\mathrm{e}^{x}+1-1\)

Simplify the expression

The terms 1 will cancel out, and we are left with just \(\mathrm{e}^{x}\). Result: \(\mathrm{e}^{x}\) (d) Simplify \(\mathrm{e}^{3 x}\left(\mathrm{e}^{-2 x}-\mathrm{e}^{-3 x}\right)+1\)

Distribute the exponent

Distribute the \(\mathrm{e}^{3 x}\) to both terms inside the parentheses. Result: \(\mathrm{e}^{3 x}\mathrm{e}^{-2 x}-\mathrm{e}^{3 x}\mathrm{e}^{-3 x}+1\)

Simplify the exponents

Using the property \(\mathrm{e}^{a}\mathrm{e}^{b}=\mathrm{e}^{a+b}\): Result: \(\mathrm{e}^{x}-\mathrm{e}^{0}+1\)

Simplify the expression

Since \(\mathrm{e}^{0}=1\), we combine the terms. Result: \(\mathrm{e}^{x}-1+1\)

Final simplified expression

The terms -1 and 1 will cancel out, and we are left with just \(\mathrm{e}^{x}\). Result: \(\mathrm{e}^{x}\)

Key concepts

Expression Simplification

Simplifying expressions is like cleaning up a messy room. You want to make everything as neat as possible before you can see the final picture. When simplifying expressions that involve exponential functions, the goal is to make the expression as compact as possible without changing its value.

Imagine you have the expression \(\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{2}+\frac{\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}\). Combining fractions is a fundamental skill. You add the numerators together while the denominator remains the same. In this case, it becomes \(\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}+\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}\).

Notice how \(\mathrm{e}^{x}\) and \(-\mathrm{e}^{-x}\) cancel each other out. This is another important part of simplification: canceling terms that are opposites of each other.

Finally, you'll end with a simplified expression where any unnecessary elements have been removed, resulting in a cleaner and more elegant form.

Exponential Laws

Exponential functions have their own set of rules known as exponential laws. These laws are crucial for handling expressions involving exponents. Let's go over some of the basic laws we can apply.

1. **Product Law**: When you multiply exponents with the same base, you add their powers: \(\mathrm{e}^{a} \cdot \mathrm{e}^{b} = \mathrm{e}^{a+b}\).

2. **Quotient Law**: When you divide exponents with the same base, you subtract the power in the denominator from the power in the numerator: \(\frac{\mathrm{e}^{a}}{\mathrm{e}^{b}} = \mathrm{e}^{a-b}\).

3. **Zero Exponent Rule**: Any non-zero base raised to the power of zero is 1: \(\mathrm{e}^{0} = 1\).

These exponential laws make manipulating expressions easier. For example, when simplifying \(\mathrm{e}^{3x} \left( \mathrm{e}^{-2x} - \mathrm{e}^{-3x} \right)\), you distribute the exponent using the product law: \(\mathrm{e}^{3x-2x}\) and \(\mathrm{e}^{3x-3x}\).

Understanding these laws will help you simplify expressions confidently and accurately.

Algebraic Manipulation

Algebraic manipulation involves using various algebraic techniques to rewrite expressions in a simpler or more useful form. This skill is particularly useful when working with complex expressions involving exponents and fractions.

One common technique is factoring. For instance, with the expression \(\frac{\mathrm{e}^{2x}+\mathrm{e}^{x}}{\mathrm{e}^{x}}-1\), you can factor out \(\mathrm{e}^{x}\) from the numerator: \(\mathrm{e}^{x}(\mathrm{e}^{x}+1)\).

Once the common factor is identified, it can be cancelled with the same factor in the denominator, leaving you with a much simpler expression. This is crucial in making burdensome calculations more manageable.

Using algebraic manipulation effectively can turn seemingly complicated expressions into simpler forms where additional operations, like addition or subtraction, can be performed easily. This approach ensures that we find the simplest form of any algebraic expression, enabling ease of further computations or applications.