Exponent Intro

Algebra-1

1. Fundamental Concepts

Definition: Exponents are a shorthand way to represent repeated multiplication of the same number.
Base and Exponent: In $$a^n$$ , $$a$$ is the base and $$n$$ is the exponent.
Zero Exponent Rule: Any non-zero number raised to the power of zero equals 1: $$a^0 = 1$$ .

2. Key Concepts

Product of Powers Rule: $$a^m \cdot a^n = a^{m+n}$$

Power of a Power Rule: $$(a^m)^n = a^{mn}$$

Quotient of Powers Rule: $$\frac{a^m}{a^n} = a^{m-n}$$

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: Calculate directly: $$2^3 = 8$$ , $$2^4 = 16$$ ; $$8 \cdot 16 = 128$$ ; Simplified: $$2^7 = 128$$ ✓

Example 2 (Intermediate)

Problem: Simplify $$\left(3^2\right)^3$$

Step-by-Step Solution:

Apply the Power of a Power Rule: $$\left(3^2\right)^3 = 3^{2 \cdot 3}$$
Simplify the exponent: $$3^6$$

Validation: Calculate directly: $$3^2 = 9$$ ; $$9^3 = 729$$ ; Simplified: $$3^6 = 729$$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use color-coding for different bases and exponents.
Error-Proofing: Double-check each step by substituting values or simplifying in parts.
Concept Reinforcement: Practice with a variety of problems that mix different rules.