Problem 10 - College Algebra with Modeling and Visualization

Question

Simplify the expression without a calculator $$ e^{3 x} e^{1+x} $$

Step-by-Step Solution

Answer

The simplified expression is \( e^{4x + 1} \).

Step 1: Apply the Properties of Exponents

The first step is to apply the property of exponents which states that when multiplying powers with the same base, you add their exponents. Here, both terms have the base \( e \). So, we can write: \[ e^{3x} imes e^{1+x} = e^{(3x) + (1+x)} \] Expand the exponents in the parenthesis by distributing the addition.

Step 2: Combine Like Terms in the Exponent

In the expression \((3x) + (1 + x)\), distribute the addition and combine like terms. Calculate: \[ (3x) + (1 + x) = 3x + 1 + x = 4x + 1 \] Now substitute back into the exponent: \[ e^{3x} imes e^{1+x} = e^{4x + 1} \]

Step 3: Write the Final Simplified Expression

Now, with the simplified exponent, you can write the entire expression. The final simplified version of the original expression is: \[ e^{4x + 1} \]

Key Concepts

Base of an Exponent

The base of an exponent is such an important component when working with powers. It is essentially the number that gets multiplied by itself. In our example of simplifying the expression \( e^{3x} e^{1+x} \), the base is \( e \). Think of the base as the foundation of a building; without it, the exponential expression wouldn't have anything to "stand on."

Key points to remember about the base:

• The base appears consistently in both terms you're multiplying, like \( e \) in this example.
• Exponents act on the base. Thus, a consistent base like \( e \) simplifies combining different parts of an expression.

A consistent base lets you apply exponential rules more easily, saving you time and avoiding confusion.

Multiplication Rule for Exponents

When you multiply exponential terms with the same base, you add their exponents. This is called the multiplication rule for exponents. It's a powerful property that can simplify otherwise complex expressions, such as \( e^{3x} e^{1+x} \).

Here's how it works:

• Identify the common base (e.g., \( e \) in our case).
• Add the exponents together, keeping the base constant. For example, \( e^{3x + (1 + x)} \) becomes \( e^{4x + 1} \).

This rule is a direct result of repeated multiplication and serves as a shortcut. You save time by skipping individual multiplications and moving straight to addition of exponents. Always ensure that the bases are indeed the same before applying this rule, as it only works when the bases match.

Exponentiation Simplification

Exponentiation simplification is all about making long expressions more manageable. Simplification techniques help in combining and reducing parts of an expression to its most compact form. Our expression started as \( e^{3x} e^{1+x} \), and through simplification, we transformed it into the more straightforward \( e^{4x + 1} \).

Steps to simplify include:

• Apply properties of exponents, such as the multiplication rule, to add exponents.
• Incorporate basic arithmetic to combine terms within the exponent. For instance, combine \( 3x + x \) into \( 4x \).
• Write the simplified expression with a single base-exponent structure wherever possible.

Simplification is key for clearer communication and easier calculation. It not only makes complex expressions easier to handle but also enhances understanding by keeping expressions clean and neat.