📚 Mastering Indices and Standard Form for IGCSE Mathematics | 精通IGCSE数学：指数与标准形式
In this revision article, we explore two core topics in IGCSE Mathematics: indices and standard form. These skills are essential for simplifying expressions, solving equations, and working with very large or very small numbers. We will explain the rules step by step, highlight common mistakes, and provide practice questions to boost your confidence.
在本文复习资料中，我们将深入探讨IGCSE数学的两个核心主题：指数与标准形式。这些技能对于简化表达式、解方程以及处理极大或极小的数字至关重要。我们将逐步解释法则、指出常见错误，并提供练习题以增强你的信心。
1. What are Indices? | 什么是指数？
An index tells us how many times a number, called the base, is multiplied by itself. For example, 2³ = 2 × 2 × 2 = 8. The number 2 is the base, and 3 is the index (or exponent).
指数告诉我们一个称为底数的数与自己相乘多少次。例如，2³ = 2 × 2 × 2 = 8。数字 2 是底数，3 是指数（或幂指数）。
In algebra, variables can also have indices, such as x⁵, y⁻², or z1/2. The same rules apply whether the base is a number or a letter.
在代数中，变量也可以带有指数，例如 x⁵、y⁻² 或 z1/2。无论底数是数字还是字母，相同的法则都适用。
2. The Multiplication Law | 乘法法则
When multiplying two powers with the same base, keep the base and add the indices.
am × an = am+n
当两个同底数的幂相乘时，底数不变，指数相加。
For example, x² × x³ = x⁵. This works because x² × x³ = (x × x) × (x × x × x) = x × x × x × x × x = x⁵.
例如，x² × x³ = x⁵。这是因为 x² × x³ = (x × x) × (x × x × x) = x × x × x × x × x = x⁵。
3. The Division Law | 除法法则
When dividing two powers with the same base, keep the base and subtract the indices.
am ÷ an = am−n
当两个同底数的幂相除时，底数不变，指数相减。
For example, y⁷ ÷ y² = y⁵. Notice that the larger index is usually written first when m is greater than n.
例如，y⁷ ÷ y² = y⁵。当 m 大于 n 时，通常先写较大的指数。
4. The Power Law | 幂的乘方法则
When raising a power to another power, multiply the indices.
(am)n = am × n
当把一个幂再乘方时，指数相乘。
For example, (3²)⁴ = 3⁸, because 2 × 4 = 8. This rule is especially useful when simplifying expressions with nested brackets.
例如，(3²)⁴ = 3⁸，因为 2 × 4 = 8。这条法则在简化含嵌套括号的表达式时特别有用。
5. Zero and Negative Indices | 零指数与负指数
Any non-zero base raised to the power zero is equal to 1. A negative index represents the reciprocal of the base raised to the positive index.
a⁰ = 1 (a ≠ 0), a−n = 1 / an
任何非零底数的零次幂都等于 1。负指数表示底数的正指数次幂的倒数。
For example, 7⁰ = 1, and 2⁻³ = 1 / 2³ = 1/8. Be careful: 2⁻³ is not −8, because the negative sign applies to the index, not the base.
例如，7⁰ = 1，2⁻³ = 1 / 2³ = 1/8。请注意：2⁻³ 不等于 −8，因为负号作用于指数，而不是底数。
6. Fractional Indices | 分数指数
A fractional index represents a root. In particular, the index 1/n means the n-th root of the base.
a1/n = ⁿ√a
分数指数表示根式。特别地，指数 1/n 表示底数的 n 次根。
More generally, am/n = (ⁿ√a)m. For example, 82/3 = (∛8)² = 2² = 4.
更一般地，am/n = (ⁿ√a)m。例如，82/3 = (∛8)² = 2² = 4。
7. Standard Form | 标准形式
Standard form, also called scientific notation, is a way of writing very large or very small numbers as a number between 1 and 10 multiplied by a power of 10.
标准形式，也称为科学记数法，是将非常大或非常小的数字写成一个介于 1 和 10 之间的数乘以 10 的幂的形式。
A × 10n, where 1 ≤ A < 10 and n is an integer
A × 10ⁿ，其中 1 ≤ A < 10，n 为整数
For example, 4 500 = 4.5 × 10³, and 0.00067 = 6.7 × 10⁻⁴. The power of 10 shows how many places the decimal point has moved.
例如，4 500 = 4.5 × 10³，0.00067 = 6.7 × 10⁻⁴。10 的幂表示小数点移动了多少位。
8. Converting to and from Standard Form | 标准形式与普通记法的转换
To convert a number to standard form, move the decimal point so that only one non-zero digit is on its left. Count the number of places moved: moving the decimal point left gives a positive power of 10, and moving it right gives a negative power.
要将数字转换为标准形式，移动小数点，使其左侧仅保留一位非零数字。统计移动的位数：小数点左移得到 10 的正幂，右移得到 10 的负
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