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스터디 가이드 - 스마트 노트
자료에 맞춘 맞춤형 노트, 핵심 정의, 예시, 맥락을 확장해 제공합니다.
Equations & Inequalities
Properties of Exponents
Understanding the properties of exponents is essential for simplifying expressions and solving equations in algebra. These rules allow us to rewrite expressions in simpler or more useful forms.
Exponent: In an expression of the form , is the base and is the exponent.
Exponential Form: Repeated multiplication can be written as ( times).
Basic Properties of Exponents
Product Rule:
Quotient Rule: ,
Power Rule:
Power of a Product Rule:
Power of a Quotient Rule: ,
Zero Exponent Rule: ,
Negative Exponent Rule: and
Rational Exponent Rule: , for even
General Rational Exponent:
Examples and Applications
Example 1: Simplify
Example 2: Rewrite as a single factor with a base of 2 , so
Example 3: Solve , so
Example 4: Simplify
Example 5: Simplify
Example 6: Solve , so
Example 7: Simplify
Special Exponent Cases
Zero Exponent: Any nonzero base raised to the zero power is 1. ,
Negative Exponent: A negative exponent indicates a reciprocal.
Fractional Exponent: A rational exponent represents a root.
General Rational Exponent:
Exponent Rules Table
Rule Name | Formula | Conditions |
|---|---|---|
Product Rule | Any | |
Quotient Rule | ||
Power Rule | Any | |
Power of a Product | Any | |
Power of a Quotient | ||
Zero Exponent | ||
Negative Exponent | ||
Rational Exponent | for even | |
General Rational Exponent | for even |
Practice Problems
Simplify:
Solve:
Simplify:
Summary
Mastering exponent rules is crucial for simplifying algebraic expressions and solving equations.
Always write final answers with positive exponents unless otherwise specified.
Be careful with negative and fractional exponents; rewrite them using roots and reciprocals as needed.
