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第 13 週 · 指數與對數 · 第 1 節

有理指數(分數指數與指數定律)

初中你會 2³ = 8;高中要把指數推廣到 0、負數、分數 —— 2⁰、2⁻¹、4^(1/2) 都是什麼?用「指數階梯」一步步講透,配滿圖例。

▶ 第一關: 指數階梯(零指數、負指數)
2³ = 2 × 2 × 2 = 8each +1 in the exponent → × 2each −1 in the exponent → ÷ 22³ = 82² = 42¹ = 22⁰ = 12⁻¹ = 1/2
aᵐ × aⁿ= aᵐ⁺ⁿaᵐ ÷ aⁿ= aᵐ⁻ⁿ(aᵐ)ⁿ= aᵐⁿ
2² = 42¹ = 22⁰ = 12⁻¹ = ½÷2÷2÷2walk down the ladder: keep dividing by 2
▶ 第二關: 分數指數 = 根式
a^(1/n)=ⁿ√a4^(1/2)=√4=28^(1/3)=³√8=216^(1/4)=2denominator = which root to take
9^(3/2)denominator 2 → √numerator 3 → cube(√9)³ = 3³ = 27rootroot first, then power = easiest
▶ 第三關: 用三定律化簡求值
2³ × 2²2^(3+2)2⁵ = 32same base → ADD the exponents
negative exponent2³ × 2⁻¹ = 2² = 4fractional exponent16^(3/4) = 2³ = 8same three laws — nothing new to fear
⚔ BOSS: 有理指數綜合
25^(3/2)(√25)³125root first (25 → 5), then cube