Problem 9 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(e^{2x}\).

Step 1: Understand the Exponential Rule

To simplify the expression that contains exponential terms, it's important to recall the property of exponents: \(a^m \cdot a^n = a^{m+n}\). This means we can combine exponents when the bases are the same.

Step 2: Apply the Exponential Rule

Given the expression \(e^x \cdot e^x\), notice the bases are the same (both \(e\)). According to the exponential rule, add the exponents: \(e^{x+x}\).

Step 3: Simplify the Exponent

Now, simplify the expression inside the exponent: \(x + x = 2x\). Thus the expression becomes \(e^{2x}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a vital skill in algebra, allowing us to handle complex mathematical statements in a more manageable form. The goal here is to reduce the expression to its simplest form without altering its value. Simplification involves using specific rules and properties to make expressions easier to understand and work with. In our example, we started with the expression \(e^x \cdot e^x\). By applying rules for exponents, we were able to simplify it to \(e^{2x}\). Simplifying such an expression often involves mathematical operations such as combining like terms or reducing fractions. In the context of exponential expressions, it typically involves using specific exponential rules to consolidate parts of the expression. Watch for similar terms and simplify them using mathematical operations as you proceed through the problem. This helps in not only making calculations easier but also in maintaining accuracy.

Exponent Rules

Exponent rules are essential tools in mathematics that help simplify expressions involving exponents. One fundamental rule is the Product of Powers property, which states that when multiplying two expressions with the same base, you can add the exponents:

• If \(a^m \cdot a^n = a^{m+n}\)

This rule is used to combine terms and make the expression simpler. For example, in our original problem, we had \(e^x \cdot e^x\). Both parts of this expression had the same base, \(e\), allowing us to add the exponents, resulting in \(e^{x+x}\) which simplifies to \(e^{2x}\). This exponent rule is crucial in algebra since it simplifies the multiplication of exponential terms considerably. When simplifying expressions, always apply the appropriate exponent rules to ensure an accurate simplification process.

Properties of Exponents

Understanding the properties of exponents is key to solving and simplifying exponential expressions efficiently. These properties outline how different operations affect expressions with exponents and make dealing with exponential expressions more manageable. In the context of our exercise:

• The Product of Powers property was utilized: \(a^m \cdot a^n = a^{m+n}\).
• This property emphasizes that when you're multiplying terms with the same base, you can just add the exponents together.

This principle is convenient in algebra, especially when working with expressions involving exponential terms. The property streamlines our calculations and forms a cornerstone of exponentiation in algebraic manipulation. Always remember to check whether bases are the same before applying exponent properties, as this is essential for accurate simplification.