Powers, Roots and Standard Form: The Story of 16³ | 幂、根与科学计数法：16³ 的故事

📚 Powers, Roots and Standard Form: The Story of 16³ | 幂、根与科学计数法：16³ 的故事

Welcome to this revision guide on powers, roots and standard form. We begin with a single number: 16³. This compact expression is packed with meaning, and unpacking it will reveal the core skills you need for your Edexcel IGCSE Maths exam.

欢迎阅读本篇关于幂、根与科学计数法的复习指南。我们从 16³ 这个简单数字开始。这个紧凑的表达式蕴含丰富内涵，拆解它将带你掌握 Edexcel IGCSE 数学考试的核心理念。

1. What Does 16³ Mean? | 16³ 的含义

The notation 16³ means 16 multiplied by itself three times. The large number 16 is called the base, and the small raised number 3 is called the index (or exponent).

记号 16³ 表示 16 自乘三次。大号数字 16 称为底数，右上角的小数字 3 称为指数（或幂）。

16³ = 16 × 16 × 16 = 4096

This simple expansion shows the power of index notation: it lets us write repeated multiplication in a compact way.

这种展开方式展示了指数记号的强大之处：它能让我们以紧凑的方式表示重复相乘。

In general, if n is a positive integer, then aⁿ means a multiplied by itself n times. For example, 2³ = 2 × 2 × 2 = 8 and 10⁵ = 100000.

一般而言，如果 n 是正整数，那么 aⁿ 表示 a 自乘 n 次。例如，2³ = 2 × 2 × 2 = 8，10⁵ = 100000。

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2. The Laws of Indices | 指数定律

One of the most powerful tools in algebra is the set of laws of indices. These rules only work when the bases are the same.

代数中最强大的工具之一就是指数定律。这些规则只在底数相同时才成立。

Rule 1: Multiplication — When multiplying powers of the same base, add the indices.

规则1：乘法 —— 当同底数幂相乘时，指数相加。

a^m × aⁿ = a^m+n

For example, 2³ × 2⁴ = 2⁷. Check: 8 × 16 = 128, and 2⁷ = 128.

例如，2³ × 2⁴ = 2⁷。验证：8 × 16 = 128，而 2⁷ = 128。

Rule 2: Division — When dividing powers of the same base, subtract the indices.

规则2：除法 —— 当同底数幂相除时，指数相减。

a^m ÷ aⁿ = a^m-n

For example, 5⁶ ÷ 5² = 5⁴ because 15625 ÷ 25 = 625, and 5⁴ = 625.

例如，5⁶ ÷ 5² = 5⁴，因为 15625 ÷ 25 = 625，而 5⁴ = 625。

Rule 3: Power of a power — Multiply the indices.

规则3：幂的乘方 —— 指数相乘。

(a^m)ⁿ = a^m×n

For example, (3²)³ = 3⁶ because 9³ = 729, and 3⁶ = 729.

例如，(3²)³ = 3⁶，因为 9³ = 729，而 3⁶ = 729。

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3. Negative and Zero Indices | 负指数与零指数

The laws of indices also work with negative indices and zero. The zero index rule says that any non-zero number raised to the power 0 equals 1.

指数定律同样适用于负指数和零指数。零指数规则规定，任何非零数的 0 次幂都等于 1。

a⁰ = 1 (a ≠ 0)

For example, 7⁰ = 1, and 163⁰ = 1.

例如，7⁰ = 1，163⁰ = 1。

A negative index indicates a reciprocal. In general:

负指数表示倒数。一般地：

a⁻ⁿ = 1 / aⁿ

So 2⁻³ = 1/8 and 10⁻² = 0.01. This rule is essential when working with standard form.

所以 2⁻³ = 1/8，10⁻² = 0.01。这条规则在处理科学计数法时至关重要。

When multiplying with negative indices, the addition rule still works: a⁻² × a⁵ = a³.

当负指数相乘时，加法规则仍然适用：a⁻² × a⁵ = a³。

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4. Fractional Indices and Roots | 分数指数与根

Fractional indices are linked to roots. The simplest cases are:

分数指数与根相关。最简单的情况是：

a^1/2 = √a, a^1/3 = ∛a

So 16^1/2 = 4 because 4² = 16, and 27^1/3 = 3 because 3³ = 27.

因此 16^1/2 = 4，因为 4² = 16；27^1/3 = 3，因为 3³ = 27。

For a more general fraction m/n, the rule is:

对于更一般的分数指数 m/n，规则是：

a^m/n = (ⁿ√a)^m or a^m/n = ⁿ√(a^m)

For example, 16^3/2 = (√16)³ = 4³ = 64. Notice that 16³ = 4096, while 16^3/2 = 64; they are different because the index is different.

例如，16^3/2 = (√16)³ =

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