Question

Simplify each expression as far as possible: (a) \(\mathrm{e}^{2 x} \mathrm{e}^{7 x}\) (b) \(\left(3 \mathrm{e}^{x}\right)\left(2 \mathrm{e}^{-x}\right)\) (c) \(\mathrm{e}^{2 x}\left(\mathrm{e}^{-2 x}+\mathrm{e}^{-x}+1\right)-\mathrm{e}^{x}\left(1+\mathrm{e}^{x}\right)\) (d) \(\frac{\mathrm{e}^{-3 x}}{2 \mathrm{e}^{-x}}\)

Short answer

Short Answer: The simplified expressions are: (a) \(\mathrm{e}^{9x}\), (b) \(6\), (c) \(1\), and (d) \(\frac{1}{2}\mathrm{e}^{-2x}\).

Step-by-step solution

Expression (a) Simplification

Given the expression \(\mathrm{e}^{2 x} \mathrm{e}^{7 x}\), apply the properties of exponents: \(\mathrm{e}^{2x} \cdot \mathrm{e}^{7x} = \mathrm{e}^{2x + 7x} = \boxed{\mathrm{e}^{9x}}\).

Expression (b) Simplification

For the expression \(\left(3 \mathrm{e}^{x}\right)\left(2 \mathrm{e}^{-x}\right)\), apply the properties of exponents and multiplication: \((3 \mathrm{e}^{x})(2 \mathrm{e}^{-x}) = 3 \cdot 2 \cdot \mathrm{e}^{x} \cdot \mathrm{e}^{-x} = 6 \cdot \mathrm{e}^{x-x} = \boxed{6}\).

Expression (c) Simplification

Simplify the expression \(\mathrm{e}^{2x}\left(\mathrm{e}^{-2x}+\mathrm{e}^{-x}+1\right)-\mathrm{e}^{x}\left(1+\mathrm{e}^{x}\right)\) as follows: 1. Distribute \(\mathrm{e}^{2x}\) across the terms within the parentheses: \(\mathrm{e}^{2x}\mathrm{e}^{-2x}+\mathrm{e}^{2x}\mathrm{e}^{-x}+\mathrm{e}^{2x}-\mathrm{e}^{x}\left(1+\mathrm{e}^{x}\right)\) 2. Apply the properties of exponents: \(1+\mathrm{e}^{x}+\mathrm{e}^{2x}-\mathrm{e}^{x}\left(1+\mathrm{e}^{x}\right)\) 3. Distribute \(-\mathrm{e}^{x}\) across the terms within the parentheses: \(1+\mathrm{e}^{x}+\mathrm{e}^{2x}-\mathrm{e}^{x}-\mathrm{e}^{2x}\) 4. Combine like terms: \(1 + \left(\mathrm{e}^{x} - \mathrm{e}^{x}\right) + \left(\mathrm{e}^{2x} - \mathrm{e}^{2x}\right) = \boxed{1}\).

Expression (d) Simplification

Finally, simplify the expression \(\frac{\mathrm{e}^{-3x}}{2\mathrm{e}^{-x}}\): 1. Apply the properties of exponents: \(\frac{\mathrm{e}^{-3x}}{2\mathrm{e}^{-x}} = \frac{1}{2} \cdot \frac{\mathrm{e}^{-3x}}{\mathrm{e}^{-x}}\). 2. Continue applying the properties of exponents: \(\frac{1}{2} \cdot \mathrm{e}^{-3x + x} = \boxed{\frac{1}{2}\mathrm{e}^{-2x}}\).

Key concepts

Properties of Exponents

When dealing with exponential functions, it's essential to understand the properties of exponents. These properties make simplifying expressions much easier.
Let's consider a couple of rules:

• Product of Powers Rule: When you multiply two exponential terms with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\). For example, \(\mathrm{e}^{2x} \times \mathrm{e}^{7x}\) simplifies to \(\mathrm{e}^{2x + 7x} = \mathrm{e}^{9x}\).
• Quotient of Powers Rule: When dividing, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\). For instance, dividing \(\mathrm{e}^{-3x}\) by \(\mathrm{e}^{-x}\) results in \(\mathrm{e}^{-3x + x} = \mathrm{e}^{-2x}\).

Using these rules correctly makes tackling exponential expressions much easier and helps in simplifying complex equations.

Simplification

Simplification in algebra involves rewriting expressions in their simplest form. This is done through careful manipulation and application of algebraic rules. Consider the example \((3 \mathrm{e}^x)(2 \mathrm{e}^{-x})\). First, multiply the constants:
3 and 2 to get 6. Next, apply the property of exponents: \(\mathrm{e}^{x} \cdot \mathrm{e}^{-x}\) simplifies to \(\mathrm{e}^{x-x} = \mathrm{e}^0 = 1\). Thus, the entire expression reduces to 6. That's the power of simplification! Reading expressions in their simplest form helps in understanding the underlying mathematical rules and detecting patterns with ease.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions or equations using algebraic rules. When you encounter an expression like \(\mathrm{e}^{2x}(\mathrm{e}^{-2x}+\mathrm{e}^{-x}+1)-\mathrm{e}^{x}(1+\mathrm{e}^{x})\), it's essential to apply logical steps:

• Distribution: Multiply out the terms to simplify. For instance, distribute \(\mathrm{e}^{2x}\) across the terms, and then do the same with \(-\mathrm{e}^{x}\).
• Combine Like Terms: After distribution, you may find terms that can cancel each other out, such as \(+\mathrm{e}^{x}\) and \(-\mathrm{e}^{x}\), or \(+\mathrm{e}^{2x}\) and \(-\mathrm{e}^{2x}\).

By carefully manipulating these terms, you can greatly simplify the expression and reach a final answer. Skilled algebraic manipulation not only simplifies expressions but also leads to a deeper understanding of how different algebraic components interact with one another.