📚 IGCSE Mathematics Revision: Surds & Indices | IGCSE数学复习：根式与指数
Surds and indices are core topics in the IGCSE Mathematics syllabus. They appear in almost every paper, from simplifying algebraic expressions to solving equations. This revision guide gives you the key rules, worked examples, and common pitfalls to avoid.
根式与指数是IGCSE数学的核心内容，几乎每张试卷都会出现，从代数式的化简到解方程都离不开它们。本复习指南将为你梳理关键法则、典型例题以及常见易错点。
1. Index Laws: The Foundation | 指数法则：基础
Indices (or powers) are shorthand for repeated multiplication. For example, x⁴ means x × x × x × x. The index laws tell us how to combine powers with the same base.
指数（或称幂）是重复乘法的简写。例如 x⁴ 表示 x × x × x × x。指数法则告诉我们如何对同底数的幂进行运算。
| Law / 法则 | Statement / 表达 | Example / 示例 |
| Multiplication / 乘法 | aᵐ × aⁿ = aᵐ⁺ⁿ | x² × x³ = x⁵ |
| Division / 除法 | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | y⁷ ÷ y² = y⁵ |
| Power of a power / 幂的幂 | (aᵐ)ⁿ = aᵐⁿ | (z²)³ = z⁶ |
| Zero index / 零指数 | a⁰ = 1 (a ≠ 0) | 5⁰ = 1 |
Always check that the bases are the same before applying the multiplication and division laws. If the bases are different, these rules do not apply directly.
在应用乘法与除法法则前，务必确保底数相同。如果底数不同，这些法则不能直接使用。
2. Negative and Zero Indices | 负指数与零指数
A negative index represents a reciprocal. The rule is a⁻ⁿ = 1/aⁿ. A zero index gives 1, provided the base is not zero.
负指数表示倒数，法则为 a⁻ⁿ = 1/aⁿ。零指数的结果为1，前提是底数不为0。
For example, 2⁻³ = 1/2³ = 1/8. Similarly, 4x⁻² = 4/x². Remember that the negative sign only applies to the index, not the base.
例如，
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