Quotient of Powers Rule

Algebra-1

1. Fundamental Concepts

Definition: The Quotient of Powers Rule states that when dividing two powers with the same base, you subtract the exponents.
Rule: For any non-zero number $a$ and integers $m$ and $n$ , $a^m \div a^n = a^{m-n}$ .
Zero Exponent Rule: Any non-zero number raised to the power of zero is 1, i.e., $a^0 = 1$ .

2. Key Concepts

Basic Rule: $$a^m \div a^n = a^{m-n}$$

Application: Used in simplifying expressions involving division of powers with the same base

Special Case: When $m = n$ , $a^m \div a^n = a^{m-m} = a^0 = 1$

3. Examples

Example 1 (Basic)

Problem: Simplify $x^5 \div x^3$

Step-by-Step Solution:

Apply the Quotient of Powers Rule: $x^5 \div x^3 = x^{5-3}$
Simplify the exponent: $x^2$

Validation: Substitute $x=2$ → Original: $2^5 \div 2^3 = 32 \div 8 = 4$ ; Simplified: $2^2 = 4$ ✓

Example 2 (Intermediate)

Problem: Simplify $\frac{y^7}{y^4}$

Step-by-Step Solution:

Apply the Quotient of Powers Rule: $\frac{y^7}{y^4} = y^{7-4}$
Simplify the exponent: $y^3$

Validation: Substitute $y=3$ → Original: $\frac{3^7}{3^4} = \frac{2187}{81} = 27$ ; Simplified: $3^3 = 27$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use color-coding for different bases and exponents to keep track of terms.
Error-Proofing: Always check if the bases are the same before applying the rule.
Concept Reinforcement: Practice with a variety of problems to reinforce understanding of the rule.