例題 1 Example 1：化簡並以正指數表示 Simplify with positive indices
化簡 (2x³y)² / (4xy⁻¹)，並以正指數表示。Simplify (2x³y)² / (4xy⁻¹) and express with positive indices.
(2x³y)² = 4x⁶y²4x⁶y² ÷ 4xy⁻¹ = x⁶⁻¹ · y²⁻⁽⁻¹⁾
答案： Answer: x⁵y³
【指數定律公式：負指數及分數指數】Laws of Indices: Index Laws, Negative and Fractional Indices
SketchMath 數學團隊 SketchMath Maths Team ｜ 更新 Updated 2026-10-20
指數定律：aᵐ × aⁿ = aᵐ⁺ⁿ、aᵐ ÷ aⁿ = aᵐ⁻ⁿ、(aᵐ)ⁿ = aᵐⁿ、(ab)ⁿ = aⁿbⁿ、a⁰ = 1、a⁻ⁿ = 1/aⁿ，以及 am/n = (ⁿ√a)ᵐ。The laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ and am/n = (ⁿ√a)ᵐ.
指數定律 Index Laws
同底相乘Multiply, same base
aᵐ × aⁿ = aᵐ⁺ⁿ同底相除Divide, same base
aᵐ ÷ aⁿ = aᵐ⁻ⁿ冪的冪Power of a power
(aᵐ)ⁿ = aᵐⁿ積及商的冪Power of a product / quotient
(ab)ⁿ = aⁿbⁿ, (a/b)ⁿ = aⁿ/bⁿ零指數及負指數Zero and negative index
a⁰ = 1, a⁻ⁿ = 1/aⁿ (a ≠ 0)分數指數Fractional index
a1/n = ⁿ√a, am/n = (ⁿ√a)ᵐ指數定律例題詳解 Worked Examples
例題 2 Example 2：指數方程 An index equation
解 4ˣ = 32。Solve 4ˣ = 32.
- 兩邊化為底數 2：
(2²)ˣ = 2⁵Write both sides with base 2:(2²)ˣ = 2⁵ 2²ˣ = 2⁵, so2x = 5
答案： Answer: x = 5/2
例題 3 Example 3：分數指數 Fractional index
求 272/3 的值。Evaluate 272/3.
271/3 = 33² = 9
答案： Answer: 9
💡 應試陷阱與秘訣 Exam Traps & Secrets
⚠️ 底數要相同 The bases must match
2³ × 3² 不能把指數相加。乘、除定律只適用於底數相同的情況。2³ × 3² cannot be combined by adding indices. The multiply and divide laws only work when the base is the same.⚠️ 負指數並非負數 Negative index is not a negative number
2⁻³ = 1/8，是正數。負號代表「倒數」。2⁻³ = 1/8, which is positive. The minus sign means "reciprocal".✏️ 自我測試：互動練習 Self-Test: Interactive Practice
溫習方法：點擊以下任何題目，即時在坐標網格載入並核對計算步驟。How to study: click any question to load it on the coordinate grid and check your steps.
(化簡 Simplify)化簡 (3a²b⁻¹)³ ÷ (9a⁻¹b²)，並以正指數表示。Simplify (3a²b⁻¹)³ ÷ (9a⁻¹b²) and express with positive indices.
(指數方程 Index equation)解 9^(x−1) = 27。Solve 9^(x−1) = 27.
(分數指數 Fractional index)求 16^(−3/4) 的值。Evaluate 16^(−3/4).
指數定律練習題（DSE 模擬題） Exercises: DSE-Style Practice
SketchMath 題庫內的 DSE 風格練習，選擇題及長題目均附解答。 DSE-style practice from the SketchMath question bank, with solutions for every MC and long question.
整數指數定律 · 中二
31 條選擇題 31 MC · 24 條長題目 24 long questions
常見問題 Frequently Asked Questions
只要底數不是 0，答案便是 1。因為 aⁿ ÷ aⁿ = a⁰ = 1。
It is 1, as long as the base is not 0. This follows from aⁿ ÷ aⁿ = a⁰ = 1.
分母是根，分子是冪：a^(m/n) = (a 的 n 次根) 的 m 次冪。
The denominator is the root and the numerator is the power: a^(m/n) = (n-th root of a) to the power m.
把兩邊化為相同底數，再令指數相等；若做不到，便取對數。
Write both sides with the same base and equate the indices. If that is impossible, take logarithms.