📚 Working with Exponents | 指数运算基础
Exponents give us a compact way to write repeated multiplication, and they are essential for simplifying expressions in algebra, geometry, and number work. This article explains what powers mean, how to evaluate them, and how to use the main exponent rules confidently in IGCSE-style questions.
指数为我们提供了一种表示重复乘法的简洁方式，对于化简代数式、几何计算和数论问题都至关重要。本文将解释幂的含义、如何求幂的值，以及如何在 IGCSE 类题目中熟练运用主要指数运算规则。
1. What Is an Exponent? | 什么是指数？
An exponent, also called an index or a power, tells you how many times to multiply a base by itself. In the expression 5³, 5 is the base and 3 is the exponent.
指数也叫次方或幂，用来说明底数要自乘多少次。在表达式 5³ 中，5 是底数，3 是指数。
5³ = 5 × 5 × 5 = 125
This means that 5³ is read as 'five to the power of three' or 'five cubed'. The exponent does not mean multiply the base by the exponent.
这意味着 5³ 读作“5 的三次方”或“5 的立方”。指数并不是把底数与指数相乘。
English: In a power, the base is the repeated factor, and the exponent counts the factors.
中文：在一个幂中，底数是重复出现的因数，指数表示因数的个数。
2. Reading Powers | 读法与含义
Powers with small exponents have special names. A number raised to the power of 2 is called squared, and a number raised to the power of 3 is called cubed.
较小指数的幂有专门名称。一个数的二次方称为平方，三次方称为立方。
6² is 'six squared'; 6³ is 'six cubed'; 6⁴ is 'six to the fourth power'.
After the third power, we usually say 'to the fourth power', 'to the fifth power', and so on. This naming helps you connect powers with area and volume units such as cm² and cm³.
三次方以后，通常读作“四次方”“五次方”等。这种命名有助于你将幂与面积、体积单位（如 cm² 和 cm³）联系起来。
3. Powers of Ten | 10 的乘方
Powers of ten are especially useful because they match our decimal place-value system. Each increase of 1 in the exponent adds one zero to the standard form of the power of ten.
10 的乘方特别有用，因为它们与十进制位值制完全对应。指数每增加 1，10 的乘方的标准形式就多一个零。
| Power | Expanded form | Value |
| 10¹ | 10 | 10 |
| 10² | 10 × 10 | 100 |
| 10³ | 10 × 10 × 10 | 1 000 |
| 10⁴ | 10 × 10 × 10 × 10 | 10 000 |
| 10⁵ | 10 × 10 × 10 × 10 × 10 | 100 000 |
Understanding powers of ten prepares you for converting large numbers and for using scientific notation at IGCSE level.
理解 10 的乘方可以为你将来在 IGCSE 学习大数转换和科学计数法打下基础。
4. Evaluating Powers | 如何求幂的值
te the base as a repeated factor and multiply. For example, 4³ means 4 × 4 × 4, not 4 × 3.
求幂的值时，先把底数写成重复因数，然后相乘。例如 4³ 表示 4 × 4 × 4，而不是 4 × 3。
4³ = 4 × 4 × 4 = 64
64. This reduces errors when powers become larger or when bases include decimals and fractions.
按步骤计算：4 × 4 = 16，再算 16 × 4 = 64。当底数包含小数或分数时，这样分步计算可以减少错误。
English: Evaluate 2⁵: 2 × 2 × 2 × 2 × 2 = 32.
中文：计算 2⁵：2 × 2 × 2 × 2 × 2 = 32。
English: Evaluate 0.5²: 0.5 × 0.5 = 0.25.
中文：计算 0.5²：0.5 × 0.5 = 0.25。
5. Zero and One as Exponents | 0 和 1 的指数
Any non-zero number raised to the power of 0 is equal to 1. This is a rule that often surprises students, but it follows from the pattern of dividing powers.
任何非零数的 0 次方都等于 1。这条规则常常让学生感到意外，但它可以通过同底数幂相除的规律推导出来。
a⁰ = 1, for a ≠ 0
A number raised to the power of 1 is simply itself. So 9¹ = 9 and x¹ = x.
任何数的 1 次方等于它本身。因此 9¹ = 9，x¹ = x。
6. Multiplying Powers with the Same Base | 同底数幂相乘
When multiplying powers that have the same base, keep the base and add the exponents. This rule works because each exponent counts the number of times the base is used as a factor.
同底数幂相乘时，底数不变，指数相加。这条规则成立的原因是：指数表示底数作为因数出现的次数。
aᵐ × aⁿ = aᵐ⁺ⁿ
⁶ because 3² is two factors of 3 and 3⁴ is four factors of 3, giving six factors in total.
示例：3² × 3⁴ = 3⁶，因为 3² 是两个 3 相乘，3⁴ 是四个 3 相乘，总共有六个 3 相乘。
3² × 3⁴ = 3⁶ = 729
Do not multiply the bases or multiply the exponents when applying this rule. The base must stay unchanged.
使用这条规则时，不要将底数相乘，也不要把指数相乘。底数必须保持不变。
7. Dividing Powers with the Same Base | 同底数幂相除
When dividing powers with the same base, keep the base and subtract the exponent of the divisor from the exponent of the dividend.
同底数幂相除时，底数不变，用被除式的指数减去除式的指数。
aᵐ ÷ aⁿ = aᵐ⁻ⁿ, for a ≠ 0
This works because dividing by 5³ removes three factors of 5 from the seven factors in 5⁷.
示例：5⁷ ÷ 5³ = 5⁴。这是因为除以 5³ 相当于从 5⁷ 的七个 5 中去掉三个 5。
5⁷ ÷ 5³ = 5⁴ = 625
re equal, the answer is 1 because the numerator and denominator divide out to give a⁰ = 1.
如果指数相等，结果为 1，因为分子和分母约分后得到 a⁰ = 1。
8. Power of a Power | 幂的乘方
When a power is raised to another power, keep the base and multiply the exponents. This is called the power of a power rule.
一个幂再乘方时，底数不变，指数相乘。这称为幂的乘方法则。
(aᵐ)ⁿ = aᵐⁿ
: (2³)² means the quantity 2³ is squared. Since 2³ = 8, we have 8² = 64, which matches 2⁶.
示例：(2³)² 表示 2³ 这个整体再平方。因为 2³ = 8，所以 8² = 64，与 2⁶ 的结果一致。
(2³)² = 2³ˣ² = 2⁶ = 64
Be careful not to add exponents here. Power of a power requires multiplication of the indices.
这里注意不要将指数相加。幂的乘方需要将指数相乘。
9. Power of a Product and Quotient | 积与商的乘方
When a product is raised to an exponent, every factor inside the parentheses is raised to that exponent.
当一个积被乘方时，括号内的每个因数都要分别乘方。
(ab)ⁿ = aⁿbⁿ
. Similarly, for a quotient, raise both the numerator and the denominator to the exponent.
示例：(3y)² = 3² × y² = 9y²。同样地，对于商的乘方，分子和分母都要分别乘方。
(a / b)ⁿ = aⁿ / bⁿ, for b ≠ 0
Example: (
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