📚 Number 6: Indices, Powers and Standard Form | 数字6：指数、幂与科学计数法
In the Edexcel IGCSE Mathematics syllabus, Number 6 focuses on the rules of indices (powers) and the use of standard form. These ideas appear throughout algebra, geometry and science, so mastering them is essential for higher marks.
在 Edexcel IGCSE 数学大纲中，数字6这一单元重点考查指数（幂）的运算规则以及科学计数法的应用。这些知识贯穿代数、几何和科学学科，掌握好它们对取得高分至关重要。
1. The Meaning of an Index | 指数的含义
An index, or power, tells you how many times a number is multiplied by itself. For example, 2³ means 2 × 2 × 2 = 8. The number 2 is called the base, and the small raised number 3 is called the index, exponent or power.
指数（index）也称为幂，表示一个数自乘的次数。例如，2³ 表示 2 × 2 × 2 = 8。其中数 2 称为底数，右上角的小数字 3 称为指数（index）、指数项（exponent）或幂（power）。
Similarly, 5² = 5 × 5 = 25, and 10⁶ = 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000. The expression a^n means a multiplied by itself n times, where n is a positive integer.
同理，5² = 5 × 5 = 25，10⁶ = 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000。表达式 a^n 表示 a 自乘 n 次，其中 n 为正整数。
2. The First Law: Multiplying Powers | 第一条法则：同底数幂相乘
When multiplying two powers with the same base, you keep the base and add the indices. For example, 3² × 3⁴ = 3^(2+4) = 3⁶.
当两个同底数幂相乘时，保留底数，指数相加。例如，3² × 3⁴ = 3^(2+4) = 3⁶。
a^m × a^n = a^(m+n)
This works because 3² × 3⁴ = (3 × 3) × (3 × 3 × 3 × 3) = 3⁶. You are simply adding the number of factor 3s.
这个法则成立是因为 3² × 3⁴ = (3 × 3) × (3 × 3 × 3 × 3) = 3⁶。你只是把因子 3 的个数相加。
- Example: x⁵ × x⁷ = x¹²
- Example: 2³ × 2⁴ = 2⁷ = 128
- 示例：x⁵ × x⁷ = x¹²
- 示例：2³ × 2⁴ = 2⁷ = 128
3. The Second Law: Dividing Powers | 第二条法则：同底数幂相除
When dividing two powers with the same base, you keep the base and subtract the indices. For example, 5⁶ ÷ 5² = 5^(6−2) = 5⁴.
当两个同底数幂相除时，保留底数，指数相减。例如，5⁶ ÷ 5² = 5^(6−2) = 5⁴。
a^m ÷ a^n = a^(m−n)
You can see this by cancelling common factors: (5 × 5 × 5 × 5 × 5 × 5) / (5 × 5) = 5 × 5 × 5 × 5 = 5⁴.
你可以通过约去公因子来理解这一法则：(5 × 5 × 5 × 5 × 5 × 5) / (5 × 5) = 5 × 5 × 5 × 5 = 5⁴。
- Example: 10⁸ ÷ 10³ = 10⁵
- Example: y⁹ ÷ y⁴ = y⁵
- 示例：10⁸ ÷ 10³ = 10⁵
- 示例：y⁹ ÷ y⁴ = y⁵
4. The Third Law: Power of a Power | 第三条法则：幂的乘方
When a power is raised to another power, you multiply the indices together. For example, (2²)³ = 2^(2×3) = 2⁶.
当一个幂再乘方时，指数相乘。例如，(2²)³ = 2^(2×3) = 2⁶。
(a^m)^n = a^(m×n)
This is because (2²)³ = 2² × 2² × 2² = 2^(2+2+2) = 2⁶.
这是因为 (2²)³ = 2² × 2² × 2² = 2^(2+2+2) = 2⁶。
- Example: (3³)² = 3⁶ = 729
- Example: (p⁴)⁵ = p²⁰
- 示例：(3³)² = 3⁶ = 729
- 示例：(p⁴)⁵ = p²⁰
5. Zero and Negative Indices | 零指数与负指数
Any non-zero number raised to the power 0 is equal to 1. For example, 7⁰ = 1 and (1/2)⁰ = 1.
任何非零数的 0 次幂都等于 1。例如，7⁰ = 1，(1/2)⁰ = 1。
a⁰ = 1 (a ≠ 0)
A negative index means the reciprocal of the power. For example, 2⁻¹ = 1/2, and 3⁻² = 1/(3²) = 1/9.
负指数表示对应正指数幂的倒数。例如，2⁻¹ = 1/2，3⁻² = 1/(3²) = 1/9。
a^(−n) = 1 / a^n (a ≠ 0)
- Example: 5⁻¹ = 1/5 = 0.2
- Example: 4⁻² = 1/16 = 0.0625
- 示例：5⁻¹ = 1/5 = 0.2
- 示例：4⁻² = 1/16 = 0.0625
6. Fractional Indices | 分数指数
A fractional index represents a root. The index 1/2 means the square root, 1/3 means the cube root, and 1/n means the nth root.
分数指数表示根式。指数 1/2 表示平方根，1/3 表示立方根，1/n 表示 n 次方根。
a^(1/n) = ⁿ√a
For example, 25^(1/2) = √25 = 5, and 8^(1/3) = ∛8 = 2.
例如，25^(1/2) = √25 = 5，8^(1/3) = ∛8 = 2。
More generally, a^(m/n) means the nth root of a, raised to the power m. For instance, 27^(2/3) = (∛27)² = 3² = 9.
更一般地，a^(m/n) 表示先对 a 开 n 次方，再取 m 次幂。例如，27^(2/3) = (∛27)² = 3² = 9。
a^(m/n) = (ⁿ√a)^m
7. Index Laws Summary | 指数法则总结
The table below summarises the key index laws you need for the IGCSE exam. Learn these thoroughly.
下表总结了 IGCSE 考试中需要掌握的关键指数法则。请务必熟记。
| Law / 法则 | Rule / 规则 | Example / 示例 |
| Multiplication | a^m × a^n = a^(m+n) | 2³ × 2² = 2⁵ |
| Division | a^m ÷ a^n = a^(m−n) | 5⁷ ÷ 5³ = 5⁴ |
| Power of a power | (a^m)^n = a^(m×n) | (3²)⁴ = 3⁸ |
| Zero index | a⁰ = 1 | 17⁰ = 1 |
| Negative index | a^(−n) = 1 / a^n | 2⁻³ = 1/8 |
| Fractional index | a^(1/n) = ⁿ√a | 16^(1/4) = 2 |
8. Standard Form: Writing Large and Small Numbers | 科学计数法：表示大数和小数
Standard form is a way of writing very large or very small numbers clearly. A number in standard form is written as A × 10^n, where 1 ≤ A < 10 and n is an integer.
科学计数法是一种清晰表示非常大或非常小的数的方法。科学计数法形式为 A × 10^n，其中 1 ≤ A < 10，n 为整数。
For example, 3,200,000 = 3.2 × 10⁶ and 0.00047 = 4.7 × 10⁻⁴.
例如，3,200,000 = 3.2 × 10⁶，0.00047 = 4.7 × 10⁻⁴。
Standard form = A × 10^n
The exponent n tells you how many places the decimal point has moved. Positive n means the original number is large; negative n means the original number is small.
指数 n 表示小数点移动的位数。n 为正表示原数较大；n 为负表示原数较小。
9. Converting Between Ordinary Numbers and Standard Form | 普通数与科学计数法的转换
To convert a large number into standard form, place the decimal point after the first non-zero digit. Count how many places the decimal point has moved; this becomes the positive power of 10.
将一个大数化为科学计数法时，把小数点放在第一个非零数字之后。记录小数点移动的位数，这个位数就是 10 的正指数。
- Example: 72,000 = 7.2 × 10⁴ (the decimal point moves 4 places left)
- Example: 1,500,000,000 = 1.5 × 10⁹
- 示例：72,000 = 7.2 × 10⁴（小数点向左移动 4 位）
- 示例：1,500,000,000 = 1.5 × 10⁹
To convert a small number into standard form, move the decimal point rightwards. The number of places moved becomes the negative power of 10.
将一个小数化为科学计数法时，小数点向右移动。移动的位数就是 10 的负指数。
- Example: 0.00035 = 3.5 × 10⁻⁴
- Example: 0.00000002 = 2 × 10⁻⁸
- 示例：0.00035 = 3.5 × 10⁻⁴
- 示例：0.00000002 = 2 × 10⁻⁸
To convert standard form back to an ordinary number, move the decimal point in the direction indicated by the power. A positive power means multiply by 10, moving the decimal point right; a negative power means divide, moving the decimal point left.
将科学计数法还原为普通数时，根据指数方向移动小数点。正指数表示乘以 10，小数点右移；负指数表示除以 10，小数点左移。
10. Working with Standard Form on the Calculator | 用计算器处理科学计数法
On most scientific calculators, you enter standard form using the × 10^x key, often labelled as EXP, EE or ×10^x. For example, to enter 4.2 × 10⁶, press 4.2, then ×10^x, then 6.
在大多数科学计算器上，输入科学计数法需要使用 × 10^x 键，通常标记为 EXP、EE 或 ×10^x。例如，要输入 4.2 × 10⁶，可按 4.2，再按 ×10^x，最后按 6。
When multiplying or dividing numbers in standard form, you can separate the number parts and the powers of 10. For example:
在计算科学计数法的乘除法时，可以将数字部分和 10 的幂分开处理。例如：
(3 × 10⁵) × (2 × 10⁴) = (3 × 2) × 10^(5+4) = 6 × 10⁹
For addition and subtraction, both numbers must first be adjusted to the same power of 10. For instance, 2.5 × 10³ + 3.1 × 10³ = 5.6 × 10³.
对于加减法，必须先将两个数化为相同的 10 的幂。例如，2.5 × 10³ + 3.1 × 10³ = 5.6 × 10³。
11. Common Mistakes and Exam Tips | 常见错误与考试提示
Many students lose marks by applying the index laws to different bases. Remember that a^m × b^n cannot be simplified unless the bases are the same.
许多学生因为对不同底数使用指数法则而失分。记住，只有底数相同时，a^m × b^n 才能化简。
Another common error is forgetting that 10⁰ = 1, or incorrectly evaluating negative powers. Negative powers do not make the answer negative; they create reciprocals.
另一个常见错误是忘记 10⁰ = 1，或错误计算负指数。负指数不表示结果为负数，而是表示倒数。
- Always check that your final answer in standard form has
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