📚 Indices & Standard Form | 指数与标准式
Indices and standard form are fundamental topics in IGCSE Mathematics. They appear in almost every paper, either as direct questions or as essential tools for solving problems in algebra, number, and even geometry. Mastering these rules will build a strong foundation for more advanced topics.
指数与标准式是 IGCSE 数学中的基础核心内容。几乎每份试卷都会出现，要么直接考察，要么作为代数、数与几何解题的重要工具。扎实掌握这些规则，将为你进一步学习更高阶内容打下坚实基础。
1. Basic Laws of Indices | 指数基本定律
The three core laws of indices are used to simplify expressions involving powers. They apply when the base number is the same.
指数的三大核心定律用于化简含幂的表达式。这些定律在底数相同时成立。
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Multiplication Law: When multiplying powers with the same base, keep the base and add the exponents.
乘法法则：同底数幂相乘，底数不变，指数相加。
aᵐ × aⁿ = aᵐ⁺ⁿ
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Division Law: When dividing powers with the same base, keep the base and subtract the exponents.
除法法则：同底数幂相除，底数不变，指数相减。
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
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Power of a Power: When raising a power to another power, multiply the exponents.
幂的乘方：指数相乘。
(aᵐ)ⁿ = aᵐⁿ
For example: 2³ × 2⁴ = 2⁷, 5⁶ ÷ 5² = 5⁴, and (3²)³ = 3⁶.
例如：2³ × 2⁴ = 2⁷，5⁶ ÷ 5² = 5⁴，(3²)³ = 3⁶。
2. Zero and Negative Indices | 零指数与负指数
These two special cases extend the meaning of powers beyond positive integers.
这两种特殊情况将幂的意义从正整数扩展到了更广的范围。
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Zero Index: Any non-zero number raised to the power 0 equals 1.
零指数：任何非零数的 0 次方都等于 1。
a⁰ = 1 (a ≠ 0)
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Negative Index: A negative exponent gives the reciprocal of the corresponding positive power.
负指数：负指数表示对应正指数幂的倒数。
a⁻ⁿ = 1 / aⁿ (a ≠ 0)
Examples: 5⁰ = 1, 2⁻¹ = ½, 10⁻³ = 1/1000, and 4⁻² = 1/16.
例如：5⁰ = 1，2⁻¹ = ½，10⁻³ = 1/1000，4⁻² = 1/16。
Notice that 2⁻³ = 1/8, not -8. The negative sign indicates the reciprocal, not a negative result.
请注意：2⁻³ = 1/8，而不是 -8。负号表示取倒数，而不是结果是负数。
3. Fractional Indices | 分数指数
Fractional exponents represent roots and combine powers with roots in a compact way.
分数指数表示开方，并将幂与根简洁地结合在一起。
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Unit Fraction Exponent: a^(1/n) is the n-th root of a.
单位分数指数：a^(1/n) 是 a 的 n 次方根。
a^(1/n) = ⁿ√a
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General Fractional Exponent: a^(m/n) means the n-th root of a, raised to the power m, or equivalently the n-th root of aᵐ.
一般分数指数：a^(m/n) 表示先对 a 开 n 次方，再取 m 次幂；也可以理解为 aᵐ 的 n 次方根。
a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
Examples: 9^(1/2) = √9 = 3, 8^(2/3) = (³√8)² = 2² = 4, and 27^(-1/3) = 1 / ³√27 = 1/3.
例如：9^(1/2) = √9 = 3，8^(2/3) = (³√8)² = 2² = 4，27^(-1/3) = 1/³√27 = 1/3。
4. Simplifying Expressions with Indices | 化简指数表达式
In examinations, you will often need to combine multiple laws to simplify algebraic expressions with variables and coefficients.
考试中，你常常需要综合运用多条定律来化简含变量与系数的代数式。
Simplify (2x³y²)⁴. Apply the power of a power law to each factor inside the bracket:
化简 (2x³y²)⁴。对括号内每一项分别应用幂的乘方法则：
(2x³y²)⁴ = 2⁴ × x¹² × y⁸ = 16x¹²y⁸
Simplify 12a⁵b³ ÷ 4a²b. Divide the coefficients and subtract the exponents for each variable:
化简 12a⁵b³ ÷ 4a²b。先除系数，再对各变量指数相减：
12a⁵b³ ÷ 4a²b = 3a³b²
When solving equations with indices, rewrite both sides with the same base where possible. For example, 2ˣ = 16 becomes 2ˣ = 2⁴, so x = 4.
解指数方程时，尽量将两边化为同底数。例如 2ˣ = 16 可写成 2ˣ = 2⁴，因此 x = 4。
5. Introduction to Standard Form | 标准式简介
Standard form (or scientific notation) is a compact way to write very large or very small numbers.
标准式（也称科学记数法）是一种书写极大或极小数的紧凑方式。
A number is in standard form when it is written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
一个数写成 A × 10ⁿ 的形式即为标准式，其中 1 ≤ A < 10，n 为整数。
N = A × 10ⁿ, 1 ≤ A < 10, n ∈ Z
For example, 4500 is written as 4.5 × 10³, not 45 × 10² or 0.45 × 10⁴, because the coefficient A must lie between 1 and 10.
例如，4500 应写成 4.5 × 10³，而不是 45 × 10² 或 0.45 × 10⁴，因为系数 A 必须介于 1 和 10 之间。
6. Converting to and from Standard Form | 与标准式的相互转换
Converting a number into standard form requires counting how many places the decimal point moves.
将数转换为标准式，需要数一数小数点移动了多少位。
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For large numbers, move the decimal point to the left until only one non-zero digit remains to its left. The number of places moved gives a positive exponent.
对于大数，将小数点向左移动，直到左边只剩一位非零数字。移动的位数就是 正 指数。
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For small numbers, move the decimal point to the right. The number of places moved gives a negative exponent.
对于小数，将小数点向右移动。移动的位数就是 负 指数。
| Number | Decimal movement | Standard form |
| 123400 | 5 places left | 1.234 × 10⁵ |
| 0.00067 | 4 places right | 6.7 × 10⁻⁴ |
| 0.00205 | 3 places right | 2.05 × 10⁻³ |
To convert from standard form back to an ordinary number, move the decimal point in the opposite direction. For example, 3.7 × 10⁴ = 37000, and 2.9 × 10⁻² = 0.029.
要从标准式转回普通数，只需沿相反方向移动小数点。例如 3.7 × 10⁴ = 37000，2.9 × 10⁻² = 0.029。
7. Calculating with Standard Form | 标准式的计算
When multiplying or dividing numbers written in standard form, handle the coefficients and the powers of 10 separately.
标准式相乘或相除时，分别处理系数与 10 的幂。
Multiplication: (A × 10ᵐ) × (B × 10ⁿ) = (A × B) × 10ᵐ⁺ⁿ.
乘法：(A × 10ᵐ) × (B × 10ⁿ) = (A × B) × 10ᵐ⁺ⁿ。
(A × 10ᵐ) × (B × 10ⁿ) = (A × B) × 10ᵐ⁺ⁿ
Division: (A × 10ᵐ) ÷ (B × 10ⁿ) = (A ÷ B) × 10ᵐ⁻ⁿ.
除法：(A × 10ᵐ) ÷ (B × 10ⁿ) = (A ÷ B) × 10ᵐ⁻ⁿ。
(A × 10ᵐ) ÷ (B × 10ⁿ) = (A ÷ B) × 10ᵐ⁻ⁿ
For addition and subtraction, first rewrite both numbers with the same exponent. For example, 3 × 10⁵ + 4 × 10⁶ is easier as 0.3 × 10⁶ + 4 × 10⁶ = 4.3 × 10⁶.
加减法时，先将两个数化为同指数。例如 3 × 10⁵ + 4 × 10⁶ 可写成 0.3 × 10⁶ + 4 × 10⁶ = 4.3 × 10⁶。
Sometimes the coefficient after a calculation is not in the range 1 ≤ A < 10. For example, (5 × 10⁴) × (4 × 10³) = 20 × 10⁷, which must be rewritten as 2 × 10⁸.
有时计算后系数不在 1 ≤ A < 10 的范围内。例如 (5 × 10⁴) × (4 × 10³) = 20 × 10⁷，必须改写为 2 × 10⁸。
8. Order of Magnitude and Estimation | 数量级与估算
Standard form is excellent for comparing the sizes of numbers and for making quick estimates.
标准式非常适合比较数字大小，也能快速进行估算。
To compare two numbers in standard form, first look at the exponent n. A larger exponent means a much larger number, regardless of the coefficient.
比较两个标准式时，先看指数 n。指数越大，数就越大，无论系数是多少。
For example, 5.1 × 10⁷ is larger than 8.9 × 10⁶, because 10⁷ is a whole order of magnitude larger than 10⁶.
例如 5.1 × 10⁷ 大于 8.9 × 10⁶，因为 10⁷ 比 10⁶ 大一个数量级。
Estimation example: Estimate (4.2 × 10⁷) × (2 × 10⁻³). Using approximate coefficients: 4.2 × 2 = 8.4, and 10⁷ × 10⁻³ = 10⁴, so the answer is about 8.4 × 10⁴.
估算例子：估算 (4.2 × 10⁷) × (2 × 10⁻³)。约算系数 4.2 × 2 = 8.4，且 10⁷ × 10⁻³ = 10⁴，所以结果约为 8.4 × 10⁴。
9. Real-World Applications | 实际应用
Standard form is widely used in science and everyday life to express very large or very small quantities.
标准式在科学和日常生活中被广泛用于表示非常大或非常小的量。
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Astronomy: The distance from the Earth to the Sun is about 1.496 × 10⁸ km.
天文学：地球到太阳的距离约为 1.496 × 10⁸ 千米。
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Physics: The charge of an electron is about 1.602 × 10⁻¹⁹ coulombs.
物理学：电子电荷量约为 1.602 × 10⁻¹⁹ 库仑。
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Biology: The length of a red blood cell is about 7 × 10⁻⁶ metres.
生物学：红细胞的直径约为 7 × 10⁻⁶ 米。
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Computer Science: A byte can represent 2⁸ = 256 different values, and data sizes are often measured in billions of bytes, e.g., 4 × 10⁹ bytes.
计算机科学：一个字节可以表示 2⁸ = 256
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