Understanding the Laws of Indices | 指数定律详解

📚 Understanding the Laws of Indices | 指数定律详解

Indices, also called powers or exponents, appear everywhere in IGCSE Mathematics. They allow us to write very large or very small numbers in a compact way, and they follow a set of simple rules. This article explains each law step by step, with worked examples and common pitfalls to avoid.

指数（也称为幂或次数）在IGCSE数学中无处不在。它们让我们能够用简洁的方式表示非常大或非常小的数字，并且遵循一套简单的规则。本文将逐步讲解每一条定律，并附有例题和常见错误提示。

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1. What Are Indices? | 什么是指数？

is called the base, and the small raised number 3 is called the index, power, or exponent.

指数告诉我们一个数字自乘的次数。例如，5³ 表示 5 × 5 × 5。数字 5 称为底数，右上角的小数字 3 称为指数、幂或次数。

5³ = 5 × 5 × 5 = 125

In general, if a is a real number and n is a positive integer, then aⁿ means a multiplied by itself n times.

一般来说，如果 a 是实数，n 是正整数，那么 aⁿ 表示 a 自乘 n 次。

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2. First Law: Multiplying Powers | 第一条定律：同底数幂相乘

When multiplying two powers with the same base, we keep the base and add the indices.

当两个同底数的幂相乘时，我们保留底数，并将指数相加。

a^m × aⁿ = a^m+n

Example: 2³ × 2⁵ = 2³⁺⁵ = 2⁸ = 256.

例如：2³ × 2⁵ = 2³⁺⁵ = 2⁸ = 256。

Why does this work? 2×2×2 and 2⁵ means 2×2×2×2×2. Multiplying them together gives eight factors of 2 in total.

为什么成立？因为 2³ 表示 2×2×2，2⁵ 表示 2×2×2×2×2。将它们相乘，总共得到 8 个 2 相乘。

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3. Second Law: Dividing Powers | 第二条定律：同底数幂相除

When dividing two powers with the same base, we keep the base and subtract the indices.

当两个同底数的幂相除时，我们保留底数，并将指数相减。

a^m ÷ aⁿ = a^m−n (a ≠ 0)

Example: 5⁶ ÷ 5² = 5⁶⁻² = 5⁴ = 625.

例如：5⁶ ÷ 5² = 5⁶⁻² = 5⁴ = 625。

Also, when the larger index is on the bottom, the result is a negative index, which we will explore later.

另外，当较大的指数在分母时，结果会出现负指数，这一点我们将在后面探讨。

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4. Third Law: Power of a Power | 第三条定律：幂的乘方

When raising a power to another power, we keep the base and multiply the indices.

当一个幂再乘方时，我们保留底数，并将指数相乘。

(a^m)ⁿ = a^m×n

Example: (3²)⁴ = 3²ˣ⁴ = 3⁸ = 6561.

例如：(3²)⁴ = 3²ˣ⁴ = 3⁸ = 6561。

Two related laws are also important:

两条相关的重要定律是：

(ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ / bⁿ

These say that a power outside a bracket applies to every factor inside the bracket.

它们说明：括号外面的指数作用于括号内的每一个因子。

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5. Zero and Negative Indices | 零指数与负指数

Any non-zero number raised to the power 0 is equal to 1.

任何非零数字的 0 次方都等于 1。

a⁰ = 1 (a ≠ 0)

A negative index represents the reciprocal of the corresponding positive power.

负指数表示对应正指数的倒数。

a⁻ⁿ = 1 / aⁿ

Example: 2⁻³ = 1 / 2³ = 1/8.

例如：2⁻³ = 1 / 2³ = 1/8。

Be careful: a negative index does not make the value negative. It only creates a fraction.

注意：负指数并不会使数值变成负数，它只是产生一个分数。

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6. Fractional Indices | 分数指数

A fractional index indicates a root. In particular, a^1/2 is the square root of a, and a^1/3 is the cube root of a.

分数指数表示开方。特别地，a^1/2 是 a 的平方根，a^1/3 是 a 的立方根。

a^1/n = ⁿ√a

Example: 9^1/2 = √9 = 3.

例如：9^1/2 = √9 = 3。

For a fractional index with numerator not equal to 1, we use:

对于分子不是 1 的分数指数，我们使用：

a^m/n = (ⁿ√a)^m

Example: 27^2/3 = (∛27)² = 3² = 9.

例如：27^2/3 = (∛27)² = 3² = 9。

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7. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Many students make the following mistakes. Read them carefully and you will save marks in the exam.

很多学生都会犯以下错误。仔细阅读它们，能帮助你在考试中避免失分。

• ting (a + b)² = a² + b². This is wrong. The correct expansion is (a + b)² = a² + 2ab + b².

  错误一：把 (a + b)² 写成 a² + b²。这是错误的。正确的展开是 (a + b)² = a² + 2ab + b²。

• Mistake 2: Confusing (a^m)ⁿ with a^maⁿ. The first gives a^mn, the second gives a^m+n.

  错误二：将 (a^m)ⁿ 与 a^maⁿ 混淆。前者结果是 a^mn，后者结果是 a^m+n。

• Mistake 3: Thinking 0⁰ = 0. In IGCSE mathematics, 0⁰ is treated as undefined.

  错误三：认为 0⁰ = 0。在IGCSE数学中，0⁰ 被认为是未定义的。

• Mistake 4: Applying index laws to addition or subtraction. For example, 2³ + 2⁵ cannot be simplified to 2⁸.

  错误四：对加法或减法使用指数定律。例如，2³ + 2⁵ 不能化简为 2⁸。

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8. Practical Applications | 实际应用

Index laws are not just abstract rules. They are used in real-life situations such as scientific notation, compound interest, and exponential growth.

指数定律不仅仅是抽象的规则。它们被应用于现实生活，例如科学记数法、复利计算和指数增长。

very large or very small numbers. For instance, the speed of light is roughly 3 × 10⁸ m/s.

科学记数法使用 10 的幂来表示非常大或非常小的数字。例如，光速大约是 3 × 10⁸ 米/秒。

Compound interest uses the formula A = P(1 + r)ⁿ, where n is the number of compounding periods. Understanding how powers behave is essential for calculating the final amount.

复利使用公式 A = P(1 + r)ⁿ，其中 n 是计息期数。理解幂的运算对于计算最终金额至关重要。

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9. Summary Table | 总结表

The table below collects all the key laws you need to remember.

下表汇总了所有需要记住的关键定律。

Law | 定律	Formula | 公式	Example | 例子
Multiplication | 乘法	am × an = am+n	2³ × 2⁴ = 2⁷
Division | 除法	am ÷ an = am−n	5⁶ ÷ 5² = 5⁴
Power of a power | 幂的乘方	(am)n = amn	(3²)⁴ = 3⁸
Zero index | 零指数	a⁰ = 1 (a≠0)	7⁰ = 1
Negative index | 负指数	a−n = 1 / an	4⁻² = 1/16
Fractional index | 分数指数	am/n = (ⁿ√a)m	272/3 = 9

Keep this summary handy. Master these laws, practise plenty of past-paper questions, and you will find that index problems become quick and easy marks.

请把这份总结放在手边。掌握这些定律，多练习历年真题，你会发现指数问题变成了快速且容易得到的分数。

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