Indices and Standard Form | 指数与科学记数法

📚 Indices and Standard Form | 指数与科学记数法

This revision guide covers two closely related topics in the IGCSE Mathematics syllabus: the laws of indices and standard form (scientific notation). These skills appear in almost every exam paper, either as direct questions or as tools for solving problems in algebra, number, and geometry. Mastering them will save you time and marks.

本复习指南涵盖IGCSE数学大纲中两个紧密相关的主题：指数的运算律和科学记数法。这些技能几乎出现在每份试卷中，要么是直接考查，要么是作为解决代数、数与几何问题的工具。掌握它们将为你节省时间并赢得分数。

1. The Meaning of an Index | 指数的含义

An index (also called an exponent or power) tells us how many times a number (the base) is multiplied by itself. For example, 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. The number 2 is the base and 5 is the index.

指数（也称为幂或次数）告诉我们一个数（底数）自乘多少次。例如，2⁵ 表示 2 × 2 × 2 × 2 × 2 = 32。其中2是底数，5是指数。

aⁿ = a × a × a × … × a (n times)

The expression aⁿ is read as ‘a to the power of n’. When the index is 2, we say ‘squared’; when it is 3, we say ‘cubed’.

表达式 aⁿ 读作“a的n次方”。当指数为2时，我们称“平方”；当指数为3时，称“立方”。

2. The First Three Laws of Indices | 指数的前三条运算法则

These three laws form the foundation of all index work. You must memorise them and know when each applies.

这三条法则构成所有指数运算的基础。你必须熟记它们，并知道每一条在何时适用。

Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ (add the indices when multiplying powers with the same base)
Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract the indices when dividing powers with the same base)
Power of a power: (aᵐ)ⁿ = aᵐⁿ (multiply the indices)

乘法：aᵐ × aⁿ = aᵐ⁺ⁿ（同底数幂相乘，指数相加）
除法：aᵐ ÷ aⁿ = aᵐ⁻ⁿ（同底数幂相除，指数相减）
幂的乘方：(aᵐ)ⁿ = aᵐⁿ（指数相乘）

Example: 3⁴ × 3² = 3⁶ = 729; 5⁷ ÷ 5³ = 5⁴ = 625; (2³)² = 2⁶ = 64

Note that these laws only work when the base is the same. You cannot combine 2³ × 3² into a single power.

注意这些法则只在底数相同时才成立。你不能将 2³ × 3² 合并成一个幂。

3. Zero and Negative Indices | 零指数与负指数

Any non-zero number raised to the power zero equals 1. For example, 7⁰ = 1 and (−4)⁰ = 1. This follows from the division law: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and since any number divided by itself is 1, a⁰ = 1.

任何非零数的零次方都等于1。例如，7⁰ = 1，(−4)⁰ = 1。这可由除法法则推出：aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰，而任何数除以自身等于1，所以a⁰ = 1。

A negative index represents a reciprocal. In general:

负指数表示倒数。一般地：

a⁻ⁿ = 1 ÷ aⁿ = 1/aⁿ

For example, 2⁻³ = 1/2³ = 1/8. A common error is to treat 2⁻³ as −8. Remember: the negative sign only affects the index, not the sign of the result.

例如，2⁻³ = 1/2³ = 1/8。一个常见错误是把 2⁻³ 当成 −8。记住：负号只影响指数，不影响结果的正负。

(2/3)⁻² = (3/2)² = 9/4

When a fraction has a negative index, take the reciprocal of the fraction first, then apply the positive index.

当分数带有负指数时，先取该分数的倒数，再应用正指数。

4. Fractional Indices | 分数指数

Fractional indices represent roots. The numerator of the fraction is the power, and the denominator is the root.

分数指数表示根式。分数的分子是幂，分母是根的次数。

a^(1/n) = ⁿ√a; a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)

In words: a to the power 1/n means the nth root of a. For example, 16^(1/2) = √16 = 4 and 8^(1/3) = ∛8 = 2.

用语言表述：a的1/n次方表示a的n次方根。例如，16^(1/2) = √16 = 4，8^(1/3) = ∛8 = 2。

Example: 27^(2/3) = (∛27)² = 3² = 9; 25^(−1/2) = 1/√25 = 1/5

Always compute the root first if it gives a whole number, as this simplifies the calculation.

如果先开根能得到整数，就应先开根，这样会简化计算。

5. Standard Form: Definition | 科学记数法的定义

Standard form (scientific notation) is a way of writing very large or very small numbers concisely. A number is in standard form when it is written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.

科学记数法是一种简洁地书写极大或极小数字的方法。当一个数写成 A × 10ⁿ 的形式时，即为科学记数法，其中 1 ≤ A < 10，n为整数。

A × 10ⁿ, where 1 ≤ A < 10

For example, 320 000 = 3.2 × 10⁵ and 0.000 47 = 4.7 × 10⁻⁴. The value of n equals the number of places the decimal point has moved.

例如，320 000 = 3.2 × 10⁵，0.000 47 = 4.7 × 10⁻⁴。n的值等于小数点移动的位数。

6. Converting Between Ordinary Numbers and Standard Form | 普通数与科学记数法的互转

To convert a large number to standard form, move the decimal point to the left until only one non-zero digit remains to its left. Count the places moved: this is n. Each place moved to the left increases n by 1.

将一个大数转换为科学记数法时，向左移动小数点，直到其左侧只剩一个非零数字。数一数移动了多少位，这就是n。向左每移动一位，n增加1。

56 000 → 5.6 × 10⁴ (decimal point moved 4 places left)

For small numbers, move the decimal point to the right until one non-zero digit remains to its left. Each place moved to the right makes n negative by 1.

对于很小的数，向右移动小数点，直到左侧只剩一个非零数字。向右每移动一位，n减小1（即负指数绝对值增加1）。

0.000 82 → 8.2 × 10⁻⁴ (decimal point moved 4 places right)

Ordinary Number	Standard Form
1 200 000	1.2 × 10⁶
0.000 003	3 × 10⁻⁶
9.4 × 10³	9400
6.5 × 10⁻²	0.065

7. Adding and Subtracting in Standard Form | 科学记数法的加减法

To add or subtract numbers in standard form, first rewrite them so that they have the same power of ten. Then add or subtract the A values, keeping the power of ten unchanged.

对科学记数法的数进行加减时，先把它们改写为相同的10的幂，然后再对A值进行加减，保持10的幂不变。

Example: (3.2 × 10⁴) + (4.5 × 10³) = (3.2 × 10⁴) + (0.45 × 10⁴) = 3.65 × 10⁴

Notice that 4.5 × 10³ was rewritten as 0.45 × 10⁴ by moving the decimal point one place left and increasing the exponent by 1. After the addition, check that the result is still in standard form: if A ≥ 10 or A < 1, adjust.

注意 4.5 × 10³ 被改写为 0.45 × 10⁴，方法是把小数点向左移一位并将指数加1。加完后，检查结果是否仍然是科学记数法：如果 A ≥ 10 或 A < 1，则需要调整。

Example: (2.8 × 10⁷) − (9.6 × 10⁶) = (2.8 × 10⁷) − (0.96 × 10⁷) = 1.84 × 10⁷

8. Multiplying and Dividing in Standard Form | 科学记数法的乘除法

For multiplication, multiply the A values together and multiply the powers of ten using the index law (add exponents). For division, divide the A values and subtract the exponents.

乘法运算中，将A值相乘，并将10的幂用指数法则相乘（指数相加）。除法运算中，将A值相除，指数相减。

(3 × 10⁵) × (2 × 10⁴) = 6 × 10⁹

(8 × 10⁷) ÷ (4 × 10³) = 2 × 10⁴

When the A values produce a result outside the range 1 ≤ A < 10, you must adjust. For example:

当A值的运算结果超出 1 ≤ A < 10 的范围时，你必须进行调整。例如：

(5 × 10³) × (6 × 10⁴) = 30 × 10⁷ = 3.0 × 10⁸

Here 30 × 10⁷ is not in standard form, so we rewrite 30 as 3.0 × 10, giving 3.0 × 10⁸.

这里 30 × 10⁷ 不是科学记数法，所以我们将30改写为 3.0 × 10，得到 3.0 × 10⁸。

9. Using a Calculator With Standard Form | 用计算器处理科学记数法

On most calculators, you enter standard form using the ×10ˣ or EXP button. For example, to enter 4.7 × 10⁻⁴, press 4.7, then the exponent button, then −4. The display often shows 4.7⁻⁰⁴ or 4.7E−4.

在大多数计算器上，使用 ×10ˣ 或 EXP 键输入科学记数法。例如，要输入 4.7 × 10⁻⁴，按4.7，再按指数键，然后按−4。显示屏通常显示 4.7⁻⁰⁴ 或 4.7E−4。

When the calculator display shows a number like 3.21E7, this means 3.21 × 10⁷. You must be able to interpret this notation in exam questions.

当计算器显示如 3.21E7 时，这表示 3.21 × 10⁷。你必须能在考试题目中解读这种表示法。

Always check whether your final answer is in standard form if the question requires it. Many exam questions explicitly say ‘Give your answer in standard form’, and you will lose a mark if you do not.

如果题目要求，一定要检查最终答案是否为科学记数法。许多考题明确要求“用科学记数法给出答案”，如果不这么做会丢分。

10. Common Exam Problems | 常见考试题型

Here are the types of questions you are most likely to encounter:

以下是你最可能遇到的题型：

Simplifying expressions: Simplify (x³)⁴ ÷ x⁵. Answer: x¹²⁻⁵ = x⁷.
Evaluating fractional powers: Find the value of 64^(2/3). Answer: (∛64)² = 4² = 16.
Converting units: A light year is 9.46 × 10¹² km. Write this as an ordinary number: 9 460 000 000 000 km.
Multiplying two numbers in standard form: (2.5 × 10⁶) × (1.4 × 10⁻²) = 3.5 × 10⁴.
Word problems involving large/small measurements: such as the mass of an atom (about 10⁻²³ g) or distances in space.

化简表达式：化简 (x³)⁴ ÷ x⁵。答案：x¹²⁻⁵ = x⁷。
计算分数指数：求 64^(2/3) 的值。答案：(∛64)² = 4² = 16。
单位换算：一光年为 9.46 × 10¹² 千米，写成普通数为 9 460 000 000 000 千米。
两个科学记数法数相乘：(2.5 × 10⁶) × (1.4 × 10⁻²) = 3.5 × 10⁴。
涉及极大/极小测量的应用题：例如原子的质量（约10⁻²³克）或太空距离。

11. Key Mistakes to Avoid | 常见错误警示

These are the top errors made by IGCSE students in examinations:

以下是IGCSE学生在考试中最常犯的错误：

Mistaking 2³ × 2² for 2⁵ (this is actually correct: 8 × 4 = 32 = 2⁵) but mixing up the law by writing 2⁶ when adding exponents of different bases.
Writing a⁻ⁿ as −aⁿ instead of 1/aⁿ.
Forgetting that a⁰ = 1 and writing a⁰ = 0.
Giving 32 × 10⁵ as a final answer instead of converting to 3.2 × 10⁶.
Confusing 10⁻³ with −10³; one is 0.001 and the other is −1000.
Forgetting to check the condition 1 ≤ A < 10 after calculations.

混淆不同底数时的指数法则，错误地把 2³ × 3² 合并成单一的幂。
把 a⁻ⁿ 写成 −aⁿ，而不是 1/aⁿ。
忘记 a⁰ = 1，错误地写成 a⁰ = 0。
最终答案给出 32 × 10⁵，而没有转换为 3.2 × 10⁶。
混淆 10⁻³ 与 −10³；前者是 0.001，后者是 −1000。
计算后忘记检查 1 ≤ A < 10 的条件。

12. Quick Revision Summary | 快速复习总结

Use this summary to check your understanding before the exam. You should be able to recall each rule without looking at the notes.

在考试前用以下总结检查你的理解。你应该能不借助笔记回忆起每一条规则。

Rule	Formula	Example
Multiplication	aᵐ × aⁿ = aᵐ⁺ⁿ	x² × x⁵ = x⁷
Division	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	y⁸ ÷ y³ = y⁵
Power of a power	(aᵐ)ⁿ = aᵐⁿ	(z²)⁴ = z⁸
Zero index	a⁰ = 1	7⁰ = 1
Negative index	a⁻ⁿ = 1/aⁿ	5⁻² = 1/25
Fractional index	a^(m/n) = (ⁿ√a)ᵐ	27^(2/3) = 9
Standard form	A × 10ⁿ, 1 ≤ A < 10	0.005 = 5 × 10⁻³

Finally, practise writing numbers in standard form without a calculator, as this strengthens your understanding of place value and exponents. Aim to complete every standard form question in past papers at least twice: once for accuracy and once for speed.

最后，练习不用计算器将数字写成科学记数法，因为这能加强你对位值和指数的理解。至少把往年试卷中的科学记数法题目做两遍：一遍求准确，一遍求速度。

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Indices and Standard Form | 指数与科学记数法