Product of Powers Rule

Algebra-1

1. Fundamental Concepts

Definition: The Product of Powers Rule states that when multiplying two powers with the same base, you add their exponents.
Base and Exponent: In the expression $$a^m$$, $$a$$ is the base and $$m$$ is the exponent.
Rule Application: For any non-zero number $$a$$ and integers $$m$$ and $$n$$, $$a^m \cdot a^n = a^{m+n}$$.

2. Key Concepts

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

Degree Preservation: The rule preserves the base while adding the exponents.

Application: Used in simplifying expressions and solving equations involving exponents.

3. Examples

Example 1 (Basic)

Problem: Simplify $$2^3 \cdot 2^4$$

Step-by-Step Solution:

Apply the Product of Powers Rule: $$2^3 \cdot 2^4 = 2^{3+4}$$
Simplify the exponent: $$2^7$$

Validation: Substitute values → Original: $$8 \cdot 16 = 128$$; Simplified: $$128$$ ✓

Example 2 (Intermediate)

Problem: Simplify $$x^5 \cdot x^{-3}$$

Step-by-Step Solution:

Apply the Product of Powers Rule: $$x^5 \cdot x^{-3} = x^{5+(-3)}$$
Simplify the exponent: $$x^2$$

Validation: Substitute $$x=2$$ → Original: $$32 \cdot \frac{1}{8} = 4$$; Simplified: $$4$$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use color-coding to distinguish different bases and exponents.
Error-Proofing: Always verify the base is the same before applying the rule.
Concept Reinforcement: Practice with a variety of bases and exponents to solidify understanding.