Power of a Power Rule

Algebra-1

1. Fundamental Concepts

Definition: The Power of a Power Rule states that raising a power to another power multiplies the exponents: $$(a^m)^n = a^{mn}$$
Base Understanding: This rule applies when the base remains the same and you are multiplying exponents.
Application: Simplifying expressions with multiple exponents on the same base.

2. Key Concepts

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

Degree Multiplication: When applying the rule, multiply the exponents directly: $$((x^3)^4) = x^{3 \cdot 4} = x^{12}$$

Application: Used in simplifying complex algebraic expressions and solving equations involving exponents

3. Examples

Example 1 (Basic)

Problem: Simplify $$(y^2)^3$$

Step-by-Step Solution:

Apply the Power of a Power Rule: $$(y^2)^3 = y^{2 \cdot 3}$$
Multiply the exponents: $$y^{6}$$

Validation: Substitute y=2 → Original: $$(2^2)^3 = 4^3 = 64$$; Simplified: $$2^6 = 64$$ ✓

Example 2 (Intermediate)

Problem: Simplify $$(3x^4)^2$$

Step-by-Step Solution:

Apply the Power of a Power Rule to the variable part: $$(x^4)^2 = x^{4 \cdot 2}$$
Multiply the exponents for the variable: $$x^8$$
Raise the coefficient to the power: $$3^2 = 9$$
Combine results: $$9x^8$$

Validation: Substitute x=1 → Original: $$(3 \cdot 1^4)^2 = 3^2 = 9$$; Simplified: $$9 \cdot 1^8 = 9$$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use color-coding to distinguish different bases and exponents.
Error-Proofing: Double-check exponent multiplication by substituting simple values.
Concept Reinforcement: Practice with a variety of bases and exponents to solidify understanding.