📚 Powers and Developments | 幂与展开式
Powers, or indices, allow us to write repeated multiplication in a compact form, while developments — the binomial expansions of powers — express expressions like (a + b)ⁿ as a sum of simpler terms. Together they underpin algebraic manipulation, series work and calculus across the Edexcel A-Level Mathematics specification.
幂（指数）让我们以紧凑的形式书写重复乘法，而展开式（development）则指将 (a + b)ⁿ 这样的幂按二项式定理展开为若干简单项之和。两者共同支撑着爱德思 A-Level 数学考试中的代数运算、级数问题与微积分内容。
This article revises the essential index laws, the binomial theorem for integer and non-integer powers, the validity conditions of infinite series, and how to apply these ideas in differentiation and integration.
本文将系统复习指数法则、整数幂与非整数幂的二项式定理、无穷级数的收敛条件，以及如何将这些思想应用于微分与积分。
1. Index Laws | 指数运算法则
An expression of the form aⁿ means the base a multiplied by itself n times. In A-Level work, the index n can be any real number: positive, negative, zero or fractional.
形如 aⁿ 的表达式表示底数 a 自乘 n 次。在 A-Level 中，指数 n 可以是任意实数：正数、负数、零或分数。
The index laws are the fundamental toolkit for simplifying powers. You must be able to recall and apply them fluently:
指数法则是化简幂运算的基本工具。你必须能够熟练地回忆并运用它们：
| Law | Rule | Example |
| Multiplication | aᵐ × aⁿ = aᵐ⁺ⁿ | x² × x⁵ = x⁷ |
| Division | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | x⁷ ÷ x² = x⁵ |
| Power of a power | Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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