IGCSE Maths: Powers, Standard Form and Estimation | IGCSE 数学：幂、标准形式与估算

📚 IGCSE Maths: Powers, Standard Form and Estimation | IGCSE 数学：幂、标准形式与估算

Welcome to this IGCSE Mathematics revision guide on powers, standard form and estimation. These skills underpin a wide range of topics, from very large astronomical distances to very small microscopic measurements. You will be expected to use index laws fluently, convert between ordinary numbers and standard form, and apply rounding or bounds in real-world problems.

欢迎阅读这篇关于幂、标准形式与估算的 IGCSE 数学复习指南。这些技能支撑着从极大的天文距离到极小的微观测量等广泛主题。考试要求你能熟练运用指数法则，在普通数与标准形式之间转换，并在实际问题中应用舍入或界限。

1. What Are Powers and Indices? | 什么是幂与指数？

A power is a short way of writing repeated multiplication. In the expression aⁿ, the base a is multiplied by itself n times. The number n is called the index, power or exponent. For example, 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Indices can be positive, zero, negative or fractional, and each type follows a clear rule.

幂是重复相乘的简写形式。在表达式 aⁿ 中，底数 a 与自身相乘 n 次。数字 n 称为指数、幂或阶。例如，2⁵ = 2 × 2 × 2 × 2 × 2 = 32。指数可以是正数、零、负数或分数，每种类型都遵循明确的规则。

2. Laws of Indices for Multiplication and Division | 指数乘除法则

When multiplying powers with the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, 3² × 3⁴ = 3²⁺⁴ = 3⁶. When dividing, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For example, 7⁵ ÷ 7² = 7⁵⁻² = 7³. These rules only work when the base is the same, so do not apply them to expressions like 2³ × 3².

同底数的幂相乘时，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。例如，3² × 3⁴ = 3²⁺⁴ = 3⁶。同底数的幂相除时，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。例如，7⁵ ÷ 7² = 7⁵⁻² = 7³。这些规则只有在底数相同时才适用，因此不要将它们用于 2³ × 3² 这类表达式。

3. Power of a Power and Zero Index | 幂的幂与零指数

When raising a power to another power, multiply the indices: (aᵐ)ⁿ = aᵐⁿ. For example, (2³)² = 2³×² = 2⁶ = 64. Any non-zero base raised to the power 0 is equal to 1: a⁰ = 1. This comes from the division rule because aᵐ ÷ aᵐ = aᵐ⁻ᵐ = a⁰, and any non-zero number divided by itself is 1.

对幂再取幂时，指数相乘：(aᵐ)ⁿ = aᵐⁿ。例如，(2³)² = 2³×² = 2⁶ = 64。任何非零底数的 0 次幂都等于 1：a⁰ = 1。这来自除法法则，因为 aᵐ ÷ aᵐ = aᵐ⁻ᵐ = a⁰，而任何非零数除以自身都等于 1。

4. Negative and Fractional Indices | 负指数与分数指数

A negative index means the reciprocal: a⁻ⁿ = 1 / aⁿ. For example, 5⁻² = 1 / 5² = 1/25. A fractional index represents a root: a 的 1/n 次幂 is the nth root of a, so ³√8 = 2 because 2³ = 8. Combining these ideas gives a^(m/n) = (ⁿ√a)ᵐ. These forms often appear when simplifying surds and solving equations.

负指数表示倒数：a⁻ⁿ = 1 / aⁿ。例如，5⁻² = 1 / 5² = 1/25。分数指数表示根：a 的 1/n 次幂是 a 的 n 次方根，因此 ³√8 = 2，因为 2³ = 8。将两者结合可得 a^(m/n) = (ⁿ√a)ᵐ。这些形式在化简根式和求解方程时经常出现。

5. Introducing Standard Form | 标准形式简介

Standard form, also called scientific notation, writes very large or very small numbers as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. For example, 6000 = 6 × 10³, and 0.0004 = 4 × 10⁻⁴. The value of n tells you how many places the decimal point moves. Positive n means a large number, and negative n means a small number.

标准形式，也称为科学记数法，将非常大或非常小的数写为 a × 10ⁿ，其中 1 ≤ a < 10，n 是整数。例如，6000 = 6 × 10³，0.0004 = 4 × 10⁻⁴。n 的值告诉你小数点移动了多少位。n 为正表示较大的数，n 为负表示较小的数。

6. Converting Between Ordinary Numbers and Standard Form | 普通数与标准形式的互换

To convert a large number into standard form, place the decimal point after the first non-zero digit and count the places moved. For 73000, write 7.3 and move the decimal point 4 places, so 7.3 × 10⁴. For a small number such as 0.0052, write 5.2 and move the decimal point 3 places to the right, so 5.2 × 10⁻³. Always check that the first part is between 1 and 10.

将较大的数转换为标准形式时，将小数点放在第一个非零数字之后，并计算移动的位数。对于 73000，写出 7.3，小数点移动了 4 位，因此为 7.3 × 10⁴。对于较小的数，如 0.0052，写出

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