Negative and Fractional Indices | 负指数与分数指数

📚 Negative and Fractional Indices | 负指数与分数指数

Understanding negative and fractional indices is fundamental to mastering algebra at A-Level. These concepts extend basic index laws, enabling you to simplify complex expressions and solve equations involving powers and roots. In this article, we will explore the rules and applications of negative exponents and rational powers, providing clear examples and common pitfalls to avoid.

理解负指数和分数指数对于掌握A-Level代数至关重要。这些概念扩展了基本的指数运算法则，使您能够简化涉及幂和根式的复杂表达式并求解方程。本文将探讨负指数和有理数幂的规则与应用，通过清晰的示例以及需要避免的常见错误，帮助您牢固掌握。

1. Review of Index Laws | 指数法则回顾

Before delving into negative and fractional indices, recall the basic index laws that hold for positive integer exponents. For any non-zero base a and integers m, n, the following rules apply:

在深入研究负指数和分数指数之前，请回顾正整指数满足的基本指数法则。对于任何非零底数 a 以及整数 m、n，下列规则成立：

• a^m × aⁿ = a^m+n | 同底数幂相乘，指数相加。
• a^m ÷ aⁿ = a^m−n | 同底数幂相除，指数相减。
• (a^m)ⁿ = a^mn | 幂的乘方，指数相乘。
• (ab)ⁿ = aⁿ bⁿ | 积的乘方等于各因式乘方的积。
• (a/b)ⁿ = aⁿ / bⁿ (b ≠ 0) | 商的乘方。

These rules form the foundation for extending indices to zero, negative, and fractional values.

这些法则是将指数扩展到零、负数和分数的基础。

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2. Zero Index | 零指数

For any non-zero number a, a⁰ = 1. This arises naturally from the division rule: a^m ÷ a^m = a^m−m = a⁰, and since any number divided by itself equals 1, we deduce a⁰ = 1. Note that 0⁰ is indeterminate.

对于任何非零数 a，a⁰ = 1。这由除法法则自然导出：a^m ÷ a^m = a^m−m = a⁰，而任何数除以自身等于1，因此可推出 a⁰ = 1。注意 0⁰ 是不确定的。

3. Negative Indices | 负指数

A negative exponent indicates the reciprocal of the base raised to the corresponding positive power: a⁻ⁿ = 1 / aⁿ (a ≠ 0). This definition is consistent with the index laws. For example, a³ × a⁻³ = a³⁻³ = a⁰ = 1, which matches if a⁻³ = 1/a³. Similarly, a fraction with a negative exponent can be flipped: (x/y)⁻ⁿ = (y/x)ⁿ.

负指数表示底数的倒数再取对应正指数幂：a⁻ⁿ = 1 / aⁿ (a ≠ 0)。此定义与指数法则一致。例如，a³ × a⁻³ = a³⁻³ = a⁰ = 1，这与 a⁻³ = 1/a³ 相符。同样，带有负指数的分数可以翻转：(x/y)⁻ⁿ = (y/x)ⁿ。

Example: Simplify 2⁻⁴ × (½)⁻³.

示例：简化 2⁻⁴ × (½)⁻³。

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