📚 Laws of Indices | 指数定律
Indices, also known as exponents or powers, provide a compact way to represent repeated multiplication of the same number. In IGCSE Mathematics, mastering the laws of indices is essential for simplifying expressions, solving equations, and working with surds and algebraic functions. This article covers all the key rules, complete with examples and typical exam applications.
指数（也称幂）提供了一种简写形式，用来表示同一个数的重复乘法。在IGCSE数学中，掌握指数定律是化简表达式、解方程以及处理根式与代数函数的基础。本文将涵盖所有关键规则，并配有例子和典型考试题型。
1. Introduction to Indices | 指数简介
An index (plural: indices) tells you how many times to use the base number in a multiplication. In the expression an, a is the base and n is the index or exponent. For example, 25 . Understanding this definition is the first step towards applying index laws confidently.
指数（单数 index，复数 indices）告诉你底数在乘法中被使用的次数。在表达式 aⁿ 中，a 是底数，n 是指数。例如，2⁵ 表示 2 × 2 × 2 × 2 × 2 = 32。理解这个定义是自信应用指数定律的第一步。
2. Positive Integer Indices | 正整数指数
When the index is a positive integer, the base is multiplied by itself that many times. The general form is an is notation works for any real base a and positive integer n. Simple evaluations include 34 = 81 and 103 = 1000. You can extend this to algebraic terms: x5 means x multiplied by itself five times.
当指数为正整数时，底数乘以自身这么多次。一般形式为 aⁿ = a × a × ... × a（n 个因子）。这个记法适用于任意实数底数 a 和正整数指数 n。简单的求值例子有 3⁴ = 81 和 10³ = 1000。这可以推广到代数项：x⁵ 表示 x 自乘五次。
3. Multiplying Powers (Product Rule) | 同底数幂相乘（乘法法则）
When multiplying two powers with the same base, you add the indices. The rule is am × an = am+n. This is because you are combining all the factors of a into one product. For instance, 42 × 43 = 42+3 = 45 = 1024. The same law applies to algebraic expressions such as p7 × p4 = p11.
同底数的两个幂相乘时，将指数相加。法则是 aᵐ × aⁿ = aᵐ⁺ⁿ。这是因为你把所有 a 的因子合并成一个乘积。例如，4² × 4³ = 4²⁺³ = 4⁵ = 1024。同样的定律适用于代数表达式，如 p⁷ × p⁴ = p¹¹。
aᵐ × aⁿ = aᵐ⁺ⁿ
4. Dividing Powers (Quotient Rule) | 同底数幂相除（除法法则）
When dividing two powers with the same base, you subtract the index of the denominator from the index of the numerator. The rule is am ÷ an = am‑n, provided a ≠ 0. For example, 79 ÷ 75 = 79‑5 = 74 = 2401. With variables, y12 / y3 simplifies to y9. Remember that the base must be the same – you cannot combine 34 and 24 in this way.
同底数的两个幂相除时，用分子的指数减去分母的指数。法则为 aᵐ ÷ aⁿ = aᵐ⁻ⁿ，前提是 a ≠ 0。例如，7⁹ ÷ 7⁵ = 7⁹⁻⁵ = 7⁴ = 2401。对于含变量的式子，y¹² ÷ y³ 化简为 y⁹。记住底数必须相同——不能把 3⁴ 和 2⁴ 这样合并。
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
5. Power of a Power Rule | 幂的乘方法则
To raise a power to another power, you multiply the indices. The rule is (am)n = am×n. This tells you that the base a is used m × n times in total. For instance, (23)2 = 23×2 = 26 = 64, which matches (8)2 = 64. Be careful with signs: (x‑2)3 = x‑6.
把一个幂再乘方时，将指数相乘。法则是 (aᵐ)ⁿ = aᵐˣⁿ。它告诉你底数 a 总共被使用了 m × n 次。例如，(2³)² = 2³ˣ² = 2⁶ = 64，这与 (8)² = 64 相符。注意符号：(x⁻²)³ = x⁻⁶。
(aᵐ)ⁿ = aᵐⁿ
6. Power of a Product | 积的乘方法则
When a product inside brackets is raised to a power, each factor in the product is raised to that power. The rule is (ab)n = an × bn. This works because (ab)n = (ab) × (ab) × ... n times = an bn. A numerical example: (2 × 5)3 = 23 × 53 = 8 × 125 = 1000. This law also extends to quotients: (a/b)n = an / bn (b ≠ 0).
当括号内的乘积被乘方时，乘积中的每个因子都要乘这个方次。法则是 (ab)ⁿ = aⁿ × bⁿ。这是因为 (ab)ⁿ 展开为 n 个 (ab) 相乘，再将 a 和 b 分别合并。数字例子：(2 × 5)³ = 2³ × 5³ = 8 × 125 = 1000。该定律也适用于商：(a/b)ⁿ = aⁿ / bⁿ （b ≠ 0）。
(ab)ⁿ = aⁿ bⁿ
7. Zero Index | 零指数
Any non‑zero number raised to the power zero is equal to 1. The rule is a0 = 1 (a ≠ 0). This can be justified using the quotient rule: am ÷ am = am‑m = a0, and we know am ÷ am = 1. Therefore, 50 = 1, (‑3)0 = 1, and (2x3)0 = 1 as long as the base is not zero. Beware: 00 is undefined.
任何非零数的零次方都等于 1。法则为 a⁰ = 1 （a ≠ 0）。这可以用除法法则来证明：aᵐ ÷ aᵐ = aᵐ⁻ᵐ = a⁰，而我们知道 aᵐ ÷ aᵐ = 1。因此，5⁰ = 1，(‑3)⁰ = 1，（2x³)⁰ = 1，只要底数不是零。切记：0⁰ 没有定义。
a⁰ = 1 (a ≠ 0)
8. Negative Indices | 负整数指数
A negative index represents the reciprocal of the positive power. The rule is a‑n = 1 / an (a ≠ 0). For example, 2‑3 = 1 / 23 = 1/8. This also works with fractions: (3/4)‑2 = (4/3)2 n with a negative exponent as its reciprocal with a positive exponent, which is very useful when simplifying algebraic fractions.
负指数表示正指数幂的倒数。法则为 a⁻ⁿ = 1 / aⁿ （a ≠ 0）。例如 2⁻³ = 1/2³ = 1/8。对于分数也适用：(3/4)⁻² = (4/3)² = 16/9。你可以把任何带有负指数的表达式改写为它的倒数并配上正指数，这在化简代数分式时非常有用。
a⁻ⁿ = 1 / aⁿ
9. Fractional Indices | 分数指数
Fractional indices combine powers and roots. The denominator of the fraction indicates the root, while the numerator indicates the power. The general rule is am/n = (ⁿ√a)m = ⁿ√am. In particular, a1/n = ⁿ√a. For example, 82/3 means the cube root of 8 squared: (³√8)2 = 22 = 4. Equivalently, 82/3 = ³√82 = ³√64 = 4. Common fractional indices are a½ = √a and a⅓ = ³√a.
分数指数将乘方与开方结合在一起。分母表示开方次数，分子表示乘方的次数。一般规则是 am/n = (ⁿ√a)ᵐ = ⁿ√aᵐ。特别地，a1/n = ⁿ√a。例如 82/3 表示 8 先平方再开立方根：(³√8)² = 2² = 4；也可写成 82/3 = ³√8² = ³√64 = 4。常见的分数指数有 a^½ = √a 以及 a^⅓ = ³√a。
am/n = (ⁿ√a)ᵐ = ⁿ√aᵐ
a^½ = √a, a^⅓ = ³√a
10. Applying Index Laws to Surds | 指数定律应用于根式
Surds are irrational roots such as √2 or ³√5. By writing them using fractional indices, you can simplify complicated expressions. For instance, √x × ³√x = x½ × x⅓ = x½+⅓ = x⁵⁄₆ = ⁶√x5. Another application is rationalising denominators: 1/√a = a‑½. Always express the final answer in its simplest surd form or as requested in the exam.
根式（surd）指的是像 √2 或 ³√5 这样的无理根。用分数指数来表示它们，你可以化简复杂的表达式。例如，√x × ³√x = x^½ × x^⅓ = x½+⅓ = x⁵⁄₆ = ⁶√x⁵。另一个应用是分母有理化：1/√a = a^⁻½。最终答案务必写成最简单的根式形式，或按考题要求作答。
11. Solving Exponential Equations | 解指数方程
When the unknown appears in the exponent, you can often solve by writing both sides with the same base. For example, to solve 2x = 32, recognise 32 = 25, so x = 5. More challenging equations require negative and fractional indices: 9x = 1/27. Write 9 = 32, 27 = 33, so (32)x = 3‑3 → 32x = 3‑3 → 2x = ‑3, giving x = ‑3/2. Always check your solution.
当未知数出现在指数位置时，通常可以通过把等式两边写成同底数来求解。例如解 2ˣ = 32，注意到 32 = 2⁵，因此 x = 5。较难的方程需要用到负指数和分数指数：9ˣ = 1/27。将 9 写成 3²，27 写成 3³，于是 (3²)ˣ = 3⁻³ → 3²ˣ = 3⁻³ → 2x = ‑3，得 x = ‑3/2。解完务必检验。
If aˣ = aⁿ, then x = n (a > 0, a ≠ 1)
12. Common Mistakes and Tips | 常见错误与提示
Misapplying the product rule: remember am × an = am+n, not am×n. For example, 23 × 24 = 27, never 212.
误用乘法法则：记住 aᵐ × aⁿ = aᵐ⁺ⁿ，而不是 aᵐˣⁿ。例如 2³ × 2⁴ = 2⁷，绝不是 2¹²。
Confusing with different bases: 23 × 53 = (10)3 is valid only because the exponents are equal, not because the bases combine directly. Use the power‑of‑a‑product rule: 23 × 53 = (2 × 5)3.
将不同底数混淆：2³ × 5³ = 10³ 之所以成立，是因为指数相同，而不是直接合并底数。要用积的乘方法则：2³ × 5³ = (2 × 5)³。
Sign errors with negative indices: x‑2 is 1/x2, not negative of x2. A negative exponent never makes the value negative – it makes it a reciprocal.
负指数符号错误：x⁻² 是 1/x²，不是 x² 的相反数。负指数绝不会让数值变负 —— 它让数变成倒数。
Forgetting that (a + b)n is not an + bn: the power‑of‑a‑sum rule does not exist in the same way. Always expand carefully using binomial techniques.
忘记 (a + b)ⁿ 不等于 aⁿ + bⁿ：没有人把和的乘方直接分配。必须用二项式展开仔细处理。
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