Laws of Indices and Standard Form | 指数法则与标准型

📚 Laws of Indices and Standard Form | 指数法则与标准型

In this revision guide we will explore the rules of indices (also called exponents or powers) and the use of standard form (scientific notation). These are core skills in the Edexcel IGCSE Mathematics syllabus, appearing in both non-calculator and calculator papers.

本复习指南将探讨指数法则（又称幂或指数）以及标准型（科学计数法）的运用。这些是爱德思 IGCSE 数学大纲中的核心技能，在非计算器和计算器试卷中都会出现。

1. Understanding Indices | 理解指数

An index (plural: indices) tells us how many times a number is multiplied by itself. For example, 5⁴ means 5 × 5 × 5 × 5.

指数表示一个数自乘的次数。例如，5⁴ 表示 5 × 5 × 5 × 5。

The number being multiplied is called the base, and the raised number is the index or exponent.

被乘的数称为底数，右上角的数称为指数或幂。

Example: In 2³, the base is 2 and the index is 3, so 2³ = 2 × 2 × 2 = 8.

例如：在 2³ 中，底数是 2，指数是 3，所以 2³ = 2 × 2 × 2 = 8。

2. The Five Basic Laws of Indices | 指数五条基本法则

These five laws allow you to simplify expressions involving powers. They only work when the bases are the same.

这五条法则能帮助化简含幂的表达式，但仅当底数相同时才适用。

Law	Rule	Example
Multiplication	aᵐ × aⁿ = aᵐ⁺ⁿ	3² × 3⁵ = 3⁷
Division	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	7⁸ ÷ 7³ = 7⁵
Power of a power	(aᵐ)ⁿ = aᵐⁿ	(2³)² = 2⁶
Power of a product	(ab)ⁿ = aⁿbⁿ	(2x)³ = 2³x³
Power of a quotient	(a/b)ⁿ = aⁿ / bⁿ	(x/2)⁴ = x⁴ / 2⁴

When multiplying, add the indices. When dividing, subtract the indices.

相乘时指数相加，相除时指数相减。

When raising a power to another power, multiply the indices.

幂的乘方时指数相乘。

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

Any non-zero base raised to the power of zero equals 1.

任何非零底数的零次幂等于 1。

a⁰ = 1 （a ≠ 0）

Example: 17⁰ = 1, and (3x)⁰ = 1, provided 3x ≠ 0.

例如：17⁰ = 1，(3x)⁰ = 1，前提是 3x ≠ 0。

A negative index represents a reciprocal.

负指数表示倒数。

a⁻ⁿ = 1 / aⁿ

Example: 2⁻³ = 1 / 2³ = 1/8; also 1 / 5⁻² = 5² = 25.

例如：2⁻³ = 1 / 2³ = 1/8；同样，1 / 5⁻² = 5² = 25。

When simplifying expressions with negative powers, flip the base and make the power positive.

化简含负幂的表达式时，将底数翻转，并使指数为正。

4. Fractional Indices | 分数指数

A fractional index indicates a root. The numerator is the power, and the denominator is the root.

分数指数表示开方。分子为幂，分母为根。

a^(1/n) = ⁿ√a

Example: 9^(1/2) = √9 = 3, and 8^(1/3) = ³√8 = 2.

例如：9^(1/2) = √9 = 3，8^(1/3) = ³√8 = 2。

For a fraction like a^(m/n), you can take the root first then raise to the power, or the other way round.

对于形如 a^(m/n) 的分数指数，可以先开方再乘方，也可以先乘方再开方。

a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)

Example: 27^(2/3) = (³√27)² = 3² = 9.

例如：27^(2/3) = (³√27)² = 3² = 9。

5. Simplifying Expressions with Indices | 化简含指数的表达式

Use the laws together to combine terms. Remember to only add or subtract indices when multiplying or dividing terms of the same base.

综合运用法则合并同类项。记住，只有在同底数相乘或相除时才能对指数进行加减。

Example: Simplify x² × x⁵ ÷ x³.

例如：化简 x² × x⁵ ÷ x³。

x² × x⁵ ÷ x³ = x^(2+5−3) = x⁴

Example: Simplify (2a³)².

例如：化简 (2a³)²。

(2a³)² = 2² × a⁶ = 4a⁶

Be careful with coefficients: square or cube the number separately from the variable part.

注意系数：对数字部分和字母部分分别乘方。

6. What Is Standard Form? | 什么是标准型？

Standard form is a way of writing very large or very small numbers as a number between 1 and 10 multiplied by a power of 10.

标准型是一种把极大或极小的数写成 1 到 10 之间的数乘以 10 的幂的形式。

A × 10ⁿ，其中 1 ≤ A < 10，n 为整数

Example: 4.5 × 10³ = 4500, and 7.2 × 10⁻² = 0.072.

例如：4.5 × 10³ = 4500，7.2 × 10⁻² = 0.072。

Notice that A is never negative in most IGCSE questions, and n can be positive or negative.

注意在大多数 IGCSE 题目中 A 不为负，n 可以是正数或负数。

7. Converting into Standard Form | 转换为标准型

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, and count how many places you moved.

将大数转换为标准型时，向左移动小数点，直到左边只剩下一个非零数字，然后数移动的位数。

Example: Write 53000 in standard form.

例如：将 53000 写成标准型。

Move the decimal point 4 places left to get 5.3. So 53000 = 5.3 × 10⁴.

小数点向左移动 4 位得到 5.3。所以 53000 = 5.3 × 10⁴。

To convert a small number, move the decimal point to the right until the first non-zero digit is the leading digit, and use a negative power.

将小数转换为标准型时，向右移动小数点，直到第一个非零数字成为首位数字，并使用负指数。

Example: Write 0.00042 in standard form.

例如：将 0.00042 写成标准型。

Move the decimal point 4 places right to get 4.2. So 0.00042 = 4.2 × 10⁻⁴.

小数点向右移动 4 位得到 4.2。所以 0.00042 = 4.2 × 10⁻⁴。

8. Calculating with Standard Form | 标准型的计算

When multiplying numbers in standard form, multiply the A parts and add the powers of 10.

标准型相乘时，将 A 部分相乘，10 的指数相加。

Example: (2 × 10³) × (3 × 10⁴) = 6 × 10⁷.

例如：(2 × 10³) × (3 × 10⁴) = 6 × 10⁷。

When dividing, divide the A parts and subtract the powers.

相除时，将 A 部分相除，指数相减。

Example: (8 × 10⁶) ÷ (2 × 10²) = 4 × 10⁴.

例如：(8 × 10⁶) ÷ (2 × 10²) = 4 × 10⁴。

When adding or subtracting, first rewrite both numbers with the same power of 10.

加减时，先把两个数改写为相同的 10 的幂。

Example: 3 × 10⁴ + 5 × 10³ = 3 × 10⁴ + 0.5 × 10⁴ = 3.5 × 10⁴.

例如：3 × 10⁴ + 5 × 10³ = 3 × 10⁴ + 0.5 × 10⁴ = 3.5 × 10⁴。

9. Adjusting the Result to Standard Form | 调整结果为标准型

Sometimes the result of a calculation is not in standard form because the A part is greater than or equal to 10.

有时计算结果并不是标准型，因为 A 部分大于或等于 10。

Example: (5 × 10⁴) × (6 × 10³) = 30 × 10⁷.

例如：(5 × 10⁴) × (6 × 10³) = 30 × 10⁷。

Since 30 is not between 1 and 10, rewrite 30 × 10⁷ as 3.0 × 10⁸.

因为 30 不在 1 到 10 之间，把 30 × 10⁷ 改写为 3.0 × 10⁸。

More generally, if A ≥ 10, move the decimal point one place left and increase the exponent by 1.

更一般地，若 A ≥ 10，将小数点左移一位，指数加 1。

If A < 1, move the decimal point one place right and decrease the exponent by 1.

若 A < 1，将小数点右移一位，指数减 1。

10. Common Exam Pitfalls | 常见考试陷阱

Many students lose marks on indices and standard form due to small but critical mistakes.

许多学生在指数和标准型上因为一些细小但关键的错误失分。

Forgetting that a⁰ = 1. 0⁰ is undefined in this syllabus.
忘记 a⁰ = 1。本大纲中 0⁰ 未定义。
Adding coefficients instead of indices when multiplying powers, e.g. 2x² × 3x³ = 6x⁵, not 5x⁶.
乘幂时把系数相加而不是指数相加，例如 2x² × 3x³ = 6x⁵，而不是 5x⁶。
Confusing the direction of moving the decimal point for negative powers.
混淆负指数时小数点的移动方向。
Using a negative exponent instead of a fractional one for roots, e.g. √x is x^(1/2), not x⁻².
对根使用负指数而不是分数指数，例如 √x 是 x^(1/2)，不是 x⁻²。
Not checking whether the final answer is in standard form.
没有检查最终答案是否为标准型。

11. Worked Exam-Style Questions | 真题风格的例题

Let us solve a few questions that are similar to Edexcel IGCSE questions.

让我们解几道类似爱德思 IGCSE 的题目。

Question 1: Simplify 4a³b² × 2ab⁵.

题目 1：化简 4a³b² × 2ab⁵。

Multiply the coefficients: 4 × 2 = 8. For a: a³ × a = a⁴. For b: b² × b⁵ = b⁷. Answer: 8a⁴b⁷.

系数相乘：4 × 2 = 8。对于 a：a³ × a = a⁴。对于 b：b² × b⁵ = b⁷。答案：8a⁴b⁷。

Question 2: Work out (4 × 10⁵) ÷ (8 × 10⁻²). Give your answer in standard form.

题目 2：计算 (4 × 10⁵) ÷ (8 × 10⁻²)，并用标准型给出答案。

Divide the A parts: 4 ÷ 8 = 0.5. Divide the powers: 10⁵ ÷ 10⁻² = 10⁽⁵⁻⁽⁻²⁾⁾ = 10⁷. So we get 0.5 × 10⁷. Since 0.5 is less than 1, rewrite as 5 × 10⁶.

将 A 部分相除：4 ÷ 8 = 0.5。指数相除：10⁵ ÷ 10⁻² = 10⁷。因此得到 0.5 × 10⁷。由于 0.5 小于 1，改写为 5 × 10⁶。

(4 × 10⁵) ÷ (8 × 10⁻²) = 5 × 10⁶

Question 3: Solve for x: 2^(x+1) = 16.

题目 3：解方程：2^(x+1) = 16。

Write 16 as 2⁴. Then x + 1 = 4, so x = 3.

将 16 写成 2⁴。于是 x + 1 = 4，所以 x = 3。

12. Final Tips | 临场建议

Memorise the laws of indices and the standard form definition before the exam.

考试前熟记指数法则和标准型的定义。

Practice moving between ordinary numbers and standard form until it feels automatic.

练习普通数与标准型之间的转换，直到形成本能。

Always show your working steps, especially when simplifying multiple terms in one expression.

始终展示你的解题步骤，特别是在化简含多项的表达式时。

Check that your final answer is in the requested form, and watch out for the difference between negative powers and fractional powers.

检查最终答案是否符合题目要求，并注意负指数与分数指数之间的区别。

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Laws of Indices and Standard Form | 指数法则与标准型