Dealing with Negative Value

Algebra-1

1. Fundamental Concepts

Definition: Exponents are a shorthand way to represent repeated multiplication of the same number.
Negative Exponents: A negative exponent indicates the reciprocal of the base raised to the positive exponent.
Rules for Exponents: When multiplying powers with the same base, add the exponents; when dividing, subtract the exponents.

2. Key Concepts

Basic Rule: $$a^{-m} = \frac{1}{a^m}$$

Degree Preservation: The rules for exponents apply regardless of whether the exponent is positive or negative.

Application: Used in various scientific and engineering calculations where small or large numbers are involved.

3. Examples

Example 1 (Basic)

Problem: Simplify $$2^{-3}$$

Step-by-Step Solution:

Apply the rule for negative exponents: $$2^{-3} = \frac{1}{2^3}$$
Calculate the denominator: $$\frac{1}{2^3} = \frac{1}{8}$$

Validation: Substitute into original expression → Original: $$2^{-3}$$; Simplified: $$\frac{1}{8}$$ ✓

Example 2 (Intermediate)

Problem: $$\frac{x^{-2}}{y^{-3}}$$

Step-by-Step Solution:

Apply the rule for negative exponents: $$\frac{x^{-2}}{y^{-3}} = \frac{\frac{1}{x^2}}{\frac{1}{y^3}}$$
Simplify the fraction: $$\frac{\frac{1}{x^2}}{\frac{1}{y^3}} = \frac{y^3}{x^2}$$

Validation: Substitute x=2, y=3 → Original: $$\frac{2^{-2}}{3^{-3}}$$; Simplified: $$\frac{3^3}{2^2} = \frac{27}{4}$$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use color-coding to distinguish between different bases and exponents.
Error-Proofing: Always check if the final answer makes sense in the context of the problem.
Concept Reinforcement: Practice with a variety of problems that involve both positive and negative exponents.