Exponent Basic Rule

Algebra-1

1. Fundamental Concepts

Definition: Exponents are a shorthand way to represent repeated multiplication of the same number.
Base and Exponent: In $$a^m$$, $$a$$ is the base and $$m$$ is the exponent.
Product Rule: When multiplying powers with the same base, add the exponents: $$a^m \cdot a^n = a^{m+n}$$

2. Key Concepts

Basic Rule: $$a^m \cdot a^n = a^{m+n}$$

Degree Preservation: The highest degree in the result matches input

Application: Used to simplify expressions in physics/engineering

3. Examples

Example 1 (Basic)

Problem: Simplify $$x^3 \cdot x^4$$

Step-by-Step Solution:

Apply the product rule: $$x^3 \cdot x^4 = x^{3+4}$$
Simplify: $$x^7$$

Validation: Substitute x=2 → Original: 8 * 16 = 128; Simplified: 128 ✓

Example 2 (Intermediate)

Problem: Simplify $$(2y^5) \cdot (3y^3)$$

Step-by-Step Solution:

Multiply coefficients: $$2 \cdot 3 = 6$$
Apply the product rule for exponents: $$y^5 \cdot y^3 = y^{5+3}$$
Combine results: $$6y^8$$

Validation: Substitute y=1 → Original: 2 * 3 * 1 * 1 = 6; Simplified: 6 * 1^8 = 6 ✓

4. Problem-Solving Techniques

Visual Strategy: Use color-coding to distinguish different bases and exponents.
Error-Proofing: Double-check that all exponents are correctly added when applying the product rule.
Concept Reinforcement: Practice with a variety of bases and exponents to solidify understanding.