Index Laws | 指数法则

📚 Index Laws | 指数法则

Index laws, also called laws of exponents, are essential tools in A-Level Mathematics. They allow you to simplify expressions involving powers, solve exponential equations, and manipulate algebraic expressions confidently. This article covers the key rules, worked examples, and common pitfalls for Edexcel A-Level Maths.

指数法则，也叫幂运算法则，是 A-Level 数学中必不可少的工具。它们能帮助你化简含有幂的表达式、解指数方程，并熟练处理代数式。本文涵盖核心法则、例题精讲和 Edexcel A-Level 数学常见错误。

1. What Are Index Laws? | 什么是指数法则？

An index (or exponent) shows how many times a base is multiplied by itself. For example, a⁵ means a × a × a × a × a. Index laws are the algebraic rules that govern how powers behave under multiplication, division, raising to another power, and taking roots.

指数（或幂）表示底数自乘的次数。例如，a⁵ 表示 a × a × a × a × a。指数法则是支配幂在乘法、除法、乘方和开方运算中如何变化的代数规则。

These laws are valid for real exponents, but in the early A-Level chapters the focus is on integer and rational exponents.

这些法则对实数指数都成立，但在 A-Level 初期章节中，重点是指数为整数和有理数的情况。

2. Multiplying Powers with the Same Base | 同底数幂相乘

When multiplying two powers with the same base, keep the base and add the exponents. This is often written as:

当两个同底数的幂相乘时，底数不变，指数相加。这通常写作：

a^m × aⁿ = a^m+n

For example, x³ × x⁵ = x⁸ and 2³ × 2⁴ = 2⁷ = 128.

例如，x³ × x⁵ = x⁸，以及 2³ × 2⁴ = 2⁷ = 128。

Always check that the bases are exactly the same. You cannot combine x² × y³ into a single power because the bases differ.

务必确认底数完全相同。你不能把 x² × y³ 合并成单个幂，因为底数不同。

3. Dividing Powers with the Same Base | 同底数幂相除

When dividing powers with the same base, subtract the exponent of the denominator from the exponent of the numerator:

当同底数的幂相除时，用分子的指数减去分母的指数：

a^m ÷ aⁿ = a^m−n

For example, x⁷ ÷ x² = x⁵ and 5⁶ ÷ 5⁴ = 5² = 25.

例如，x⁷ ÷ x² = x⁵，以及 5⁶ ÷ 5⁴ = 5² = 25。

This rule is also why a⁰ = 1: if m = n, then a^m ÷ a^m = a^m−m = a⁰, and the original quotient equals 1.

这条法则也解释了为什么 a⁰ = 1：如果 m = n，那么 a^m ÷ a^m = a^m−m = a⁰，而原始商等于 1。

4. Power of a Power | 幂的乘方

When a power is raised to another power, multiply the exponents:

当一个幂再被乘方时，指数相乘：

(a^m)ⁿ = a^mn

For example, (x³)⁴ = x¹² and (2²)³ = 2⁶ = 64

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