Indices and Standard Form | 指数与标准形式

📚 Indices and Standard Form | 指数与标准形式

In IGCSE Mathematics, mastering indices (powers) and standard form is essential for simplifying algebraic expressions and representing very large or very small numbers. This article explains the rules step by step, with worked examples and common pitfalls.

在 IGCSE 数学中，掌握指数（幂）和标准形式是化简代数表达式、表示极大或极小数字的关键。本文将逐步讲解规则、例题和常见错误。

1. What Are Indices? | 什么是指数？

An index (plural: indices) tells us how many times a number is multiplied by itself. For example, 5³ means 5 × 5 × 5 = 125. The number 5 is the base, and 3 is the index.

指数告诉我们一个数自乘多少次。例如，5³ 表示 5 × 5 × 5 = 125。数字 5 称为底数，3 称为指数。

aⁿ = a × a × … × a (n times), where n is a positive integer.
aⁿ = a × a × … × a（共 n 个 a），其中 n 为正整数。

When the index is 1, a¹ = a. When the index is 0, a⁰ = 1 (provided a ≠ 0).

当指数为 1 时，a¹ = a。当指数为 0 时，a⁰ = 1（前提是 a ≠ 0）。

2. Multiplication Law | 乘法法则

When multiplying two powers with the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ.

当同底数幂相乘时，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。

3² × 3⁴ = 3²⁺⁴ = 3⁶ = 729

This works because 3² × 3⁴ = (3×3) × (3×3×3×3) = 3⁶.

这是因为 3² × 3⁴ 展开后共有 6 个 3 相乘。

3. Division Law | 除法法则

When dividing two powers with the same base, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0).

当同底数幂相除时，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ（a ≠ 0）。

5⁷ ÷ 5² = 5⁷⁻² = 5⁵ = 3125

For example, 5⁷ ÷ 5² means (5×5×5×5×5×5×5) ÷ (5×5), leaving 5⁵.

例如，5⁷ ÷ 5² 表示分子有 7 个 5，分母有 2 个 5，约分后剩下 5 个 5。

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

A non-zero base raised to the zero power equals 1: a⁰ = 1.

任何非零底数的零次幂都等于 1：a⁰ = 1。

A negative index means the reciprocal: a⁻ⁿ = 1/aⁿ.

负指数表示倒数：a⁻ⁿ = 1/aⁿ。

7⁰ = 1
2⁻³ = 1/2³ = 1/8
(2/3)⁻² = (3/2)² = 9/4

Be careful: negative indices do not make the number negative; they make it a fraction.

注意：负指数不会让结果变成负数，而是变成分数。

5. Fractional Indices | 分数指数

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

分数指数表示根式。分母是根的次数，分子是幂。

a¹ᐟ² = √a
a¹ᐟ³ = ∛a
aᵐᐟⁿ = (ⁿ√a)ᵐ = ⁿ√(aᵐ)

8²ᐟ³ = (∛8)² = 2² = 4

Always simplify the root first if possible. This reduces the size of the numbers.

如果可能，先求根再乘方，这样数字会小很多。

6. Power of a Power | 幂的乘方

When raising a power to another power, multiply the indices: (aᵐ)ⁿ = aᵐⁿ.

幂的乘方时，指数相乘：(aᵐ)ⁿ = aᵐⁿ。

(4³)² = 4³ˣ² = 4⁶ = 4096

This also applies to products and quotients: (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ.

该法则也适用于积和商：(ab)ⁿ = aⁿbⁿ，且 (a/b)ⁿ = aⁿ/bⁿ。

7. Standard Form: Large Numbers | 标准形式：大数字

Standard form writes numbers as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. For large numbers, n is positive.

标准形式把数字写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。对于大数字，n 是正数。

45 000 = 4.5 × 10⁴

To convert, move the decimal point so that only one non-zero digit remains before it. The number of places moved gives n.

转换时，移动小数点到第一位非零数字之后。移动的位数就是 n。

8. Standard Form: Small Numbers | 标准形式：小数字

For very small numbers, n is negative. For example, 0.0007 = 7 × 10⁻⁴.

对于非常小的数字，n 是负数。例如，0.0007 = 7 × 10⁻⁴。

Count the number of places the decimal point moves to the right until A is between 1 and 10. This count becomes the negative index.

计算小数点向右移动到 A 在 1 到 10 之间需要移动的位数，这个位数就是负指数。

0.000 025 = 2.5 × 10⁻⁵
0.000 000 103 = 1.03 × 10⁻⁷

Note that leading zeros are not counted; we count the position after the first non-zero digit.

注意：前面的零不计入，我们数的是第一个非零数字之后的位置。

9. Arithmetic in Standard Form | 标准形式的运算

To multiply in standard form, multiply the A parts and add the powers of 10.

标准形式相乘时，将 A 部分相乘，10 的指数相加。

(3 × 10⁵) × (2 × 10²) = 6 × 10⁷

For division, divide the A parts and subtract the powers.

相除时，将 A 部分相除，10 的指数相减。

(8 × 10⁶) ÷ (2 × 10²) = 4 × 10⁴

When adding or subtracting, convert to the same power of 10 first. Always present the final answer in correct standard form: A must be between 1 and 10.

加减运算时，先统一成相同的 10 的幂。最终结果必须满足 A 在 1 到 10 之间。

10. Common Mistakes | 常见错误

One common error is writing 10² × 10³ = 10⁶, but the correct answer is 10⁵ because you add indices, not multiply them.

常见错误之一是写出 10² × 10³ = 10⁶，正确答案是 10⁵，因为指数应相加而非相乘。

Another mistake is misusing zero index: 0⁰ is not defined in IGCSE. Also, a⁻ⁿ is not negative; it is a reciprocal.

另一个错误是误用零指数：IGCSE 中 0⁰ 未定义。并且 a⁻ⁿ 不是负数，而是倒数。

When converting to standard form, ensure A is at least 1 and less than 10. For example, 12 × 10⁴ is not standard form; it should be 1.2 × 10⁵.

转换为标准形式时，确保 A 大于等于 1 且小于 10。例如，12 × 10⁴ 不是标准形式，应写成 1.2 × 10⁵。

11. Exam Tips | 考试提示

Always simplify expressions using index laws before evaluating. For standard form, check the sign and the value of n carefully.

先用指数法则化简表达式再求值。对于标准形式，仔细检查 n 的正负和数值。

Write down every step to avoid careless errors.
把每一步写出来，避免粗心错误。
Practise converting between index form and expanded form.
练习指数形式与展开形式之间的转换。
Use a calculator only to check, not to replace understanding.
仅用计算器自检，不要用它替代理解。

If you get an answer like 0.5 × 10⁻³, convert it to 5 × 10⁻⁴ before writing the final answer.

如果你得到类似 0.5 × 10⁻³ 的结果，在写出最终答案前要转换为 5 × 10⁻⁴。

12. Practice Questions | 练习题目

Try these questions on your own, then check your answers.

请先独立尝试以下题目，再核对答案。

Simplify 2³ × 2⁴.
Simplify 5⁶ ÷ 5².
Evaluate (3²)³.
Write 0.000 056 in standard form.
Calculate (4 × 10⁻²) × (2 × 10⁵) and give the answer in standard form.

Answers: 1. 2⁷ = 128 2. 5⁴ = 625 3. 3⁶ = 729 4. 5.6 × 10⁻⁵ 5. 8 × 10³

Be sure to review any incorrect answers and identify which step was wrong. Repetition is the key to mastering indices.

请务必回顾错题，找出出错步骤。反复练习是掌握指数的关键。

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Indices and Standard Form | 指数与标准形式