Laws of Indices | 指数法则

📚 Laws of Indices | 指数法则

Indices (or exponents) are a fundamental tool in GCSE and IGCSE mathematics. Mastering the laws of indices allows you to simplify expressions, solve equations, and work confidently with powers in every area of algebra.

指数（或称幂）是 GCSE 和 IGCSE 数学中的基础工具。掌握指数法则，你就能简化表达式、求解方程，并在代数的各个领域自信地处理幂运算。

1. What is an Index? | 什么是指数？

An index (plural: indices) tells us how many times a number or variable is multiplied by itself. In the expression \(5^3\), the base is 5 and the index is 3, meaning \(5 \times 5 \times 5\).

指数（复数：indices）表示一个数或变量自乘的次数。在表达式 5³ 中，底数是 5，指数是 3，表示 5 × 5 × 5。

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

The number being raised to a power is called the base, and the power itself is the index. For example, in \(x^7\), \(x\) is the base and 7 is the index.

被乘方的数称为底数，右上角的数称为指数。例如，在 x⁷ 中，x 是底数，7 是指数。

2. The Multiplication Law | 乘法法则

When multiplying two powers with the same base, keep the base and add the indices.

当两个同底数的幂相乘时，保留底数，指数相加。

a^m × aⁿ = a^m+n

For example, \(x^3 \times x^4 = x^{3+4} = x^7\). This works because \(x^3 = x \times x \times x\) and \(x^4 = x \times x \times x \times x\), giving seven factors in total.

例如，x³ × x⁴ = x³⁺⁴ = x⁷。这是因为 x³ = x·x·x，x⁴ = x·x·x·x，总共七个因子相乘。

\(2^3 \times 2^5 = 2^8 = 256\)
\(y^2 \times y = y^3\) (remember: \(y = y^1\))

2³ × 2⁵ = 2⁸ = 256
y² × y = y³（记住：y = y¹）

3. The Division Law | 除法法则

When dividing two powers with the same base, subtract the indices.

当两个同底数的幂相除时，保留底数，指数相减。

a^m ÷ aⁿ = a^m−n

For example, \(\frac{x^7}{x^2} = x^{7-2} = x^5\). This is because two of the seven factors cancel out.

例如，x⁷ ÷ x² = x⁷⁻² = x⁵。这是因为七个因子中有两个相互抵消。

\(5^6 \div 5^3 = 5^3 = 125\)
\(\frac{a^8}{a} = a^7\) since \(a = a^1\)

5⁶ ÷ 5³ = 5³ = 125
a⁸ ÷ a = a⁷，因为 a = a¹

4. The Power of a Power | 幂的乘方

When raising a power to another power, multiply the indices.

当一个幂再乘方时，指数相乘。

(a^m)ⁿ = a^{m × n}

For example, \((x^2)^3 = x^{2 \times 3} = x^6\). The outer index 3 means the whole expression \(x^2\) is used three times: \(x^2 \times x^2 \times x^2\).

例如，(x²)³ = x²ˣ³ = x⁶。外层指数 3 表示整个 x² 被使用三次：x² × x² × x²。

Be careful with signs: \((−2)^3 = −8\), but \((−2)^2 = 4\). The parentheses matter!

注意符号：(−2)³ = −8，但 (−2)² = 4。括号很重要！

5. Zero Index | 零指数

Any non-zero number raised to the power 0 equals 1.

任何非零数的 0 次幂都等于 1。

a⁰ = 1 (a ≠ 0)

Why? Consider \(a^m \div a^m = 1\). Using the division law, \(a^{m-m} = a^0\). So \(a^0 = 1\).

为什么？考虑 aᵐ ÷ aᵐ = 1。根据除法法则，aᵐ⁻ᵐ = a⁰。因此 a⁰ = 1。

\(7^0 = 1\)
(−3)⁰ = 1
\(x^0 = 1\) (provided \(x \ne 0\))

7⁰ = 1
(−3)⁰ = 1
x⁰ = 1（前提 x ≠ 0）

6. Negative Indices | 负指数

A negative index represents the reciprocal of the base raised to the positive index.

负指数表示底数取正指数后的倒数。

a⁻ⁿ = 1 ÷ aⁿ = 1 / aⁿ

For example, \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\). This follows from the division law: \(a^0 ÷ a^n = a^{0-n} = a^{-n}\), and \(a^0 ÷ a^n = 1/a^n\).

例如，2⁻³ = 1/2³ = 1/8。这可由除法法则推得：a⁰ ÷ aⁿ = a⁰⁻ⁿ = a⁻ⁿ，而 a⁰ ÷ aⁿ = 1/aⁿ。

\(5^{-1} = \frac{1}{5}\)
\(\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}\)

5⁻¹ = 1/5
(2/3)⁻² = (3/2)² = 9/4

7. Fractional Indices | 分数指数

Fractional indices represent roots. The denominator of the fraction is the root, and the numerator is the power.

分数指数表示根式。分母表示开方次数，分子表示幂次。

a^1/n = ⁿ√a , a^m/n = (ⁿ√a)^m = ⁿ√(a^m)

For example, \(9^{1/2} = \sqrt{9} = 3\), and \(8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4\).

例如，9¹ᐟ² = √9 = 3，8²ᐟ³ = (∛8)² = 2² = 4。

\(x^{1/3} = \sqrt[3]{x}\)
\(16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8\)
\(25^{-1/2} = \frac{1}{\sqrt{25}} = \frac{1}{5}\)

x¹ᐟ³ = ∛x
16³ᐟ⁴ = (⁴√16)³ = 2³ = 8
25⁻¹ᐟ² = 1/√25 = 1/5

8. Scientific Notation and Indices | 科学计数法与指数

Scientific notation uses powers of 10 to write very large or very small numbers. A number in scientific notation is written as \(A \times 10^n\), where \(1 \le A < 10\) and \(n\) is an integer.

科学计数法使用 10 的幂来表示很大或很小的数。科学计数法的形式为 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。

Number / 数字	Scientific notation / 科学计数法
4500	4.5 × 10³
0.00072	7.2 × 10⁻⁴

When multiplying numbers in scientific notation, multiply the coefficients and add the exponents. For example, \((3 \times 10^4) \times (2 \times 10^6) = 6 \times 10^{10}\).

科学计数法相乘时，系数相乘，指数相加。例如：(3 × 10⁴) × (2 × 10⁶) = 6 × 10¹⁰。

9. Simplifying Expressions with Indices | 含指数的表达式化简

In algebra, you often need to simplify expressions that involve letters raised to powers. Apply the laws step by step.

在代数中，你经常需要化简含字母幂的表达式。逐步应用法则即可。

Simplify \( (2x^3 y^2)^4 \).

化简 (2x³y²)⁴。

(2x³y²)⁴ = 2⁴ × x³ˣ⁴ × y²ˣ⁴ = 16x¹²y⁸

Simplify \(\frac{6x^5}{2x^2}\). First divide the coefficients: \(6 \div 2 = 3\). Then subtract indices: \(x^{5-2}=x^3\). So the answer is \(3x^3\).

化简 6x⁵ ÷ 2x²。先除系数：6 ÷ 2 = 3。再减指数：x⁵⁻² = x³。因此答案是 3x³。

10. Common Mistakes | 常见错误

Many students make the same mistakes when working with indices. Be alert for these pitfalls.

许多学生在处理指数时都会犯同样的错误。请警惕这些陷阱。

Adding exponents when bases are different: \(2^3 \times 3^2 \neq 6^5\). You may only add exponents if the base is exactly the same.
Forgetting that parentheses yield the power for the whole base: \((−3)^2 = 9\), but \(−3^2 = −9\).
Confusing \(2^3\) with \(3^2\) or \(2 \times 3\). \(2^3 = 8\), not 6.
Writing \(a^0 = 0\) instead of \(a^0 = 1\).

底数不同时相加指数：2³ × 3² ≠ 6⁵。只有底数完全相同时才能相加指数。
忘记括号意味着指数作用于整个底数：(−3)² = 9，而 −3² = −9。
混淆 2³ 与 3² 或 2 × 3。2³ = 8，不是 6。
把 a⁰ 写为 0，实际上 a⁰ = 1。

11. Solving Equations with Indices | 指数方程求解

To solve equations such as \(2^x = 8\), express both sides with the same base. Since \(8 = 2^3\), we get \(x = 3\).

求解像 2ˣ = 8 这样的方程，需要将两边化为同底数。因为 8 = 2³，所以 x = 3。

For more complex equations, use the substitution method. Solve \(4^x = 32\). Write \(4 = 2^2\) and \(32 = 2^5\), giving \((2^2)^x = 2^5\), hence \(2^{2x} = 2^5\), so \(2x = 5\), \(x = 2.5\).

对于更复杂的方程，可用换底法。解 4ˣ = 32。写成 4 = 2²，32 = 2⁵，得 (2²)ˣ = 2⁵，因此 2²ˣ = 2⁵，于是 2x = 5，x = 2.5。

12. Exam Tip: Work Backwards | 考试技巧：逆推

In the exam, you may be asked to find a missing index. For example: find \(n\) if \(3^n = 27\). Recognise that \(27 = 3^3\), so \(n = 3\).

考试中可能会要求你求缺失的指数。例如：若 3ⁿ = 27，求 n。识别 27 = 3³，所以 n = 3。

If the answer is a fraction, do not panic. Convert to fractional exponents: \(\sqrt{x} = x^{1/2}\), \(\frac{1}{x^2} = x^{-2}\), and \(\sqrt[3]{x^2} = x^{2/3}\).

如果答案是分数，不要慌张。学会转换为分数指数：√x = x¹ᐟ²，1/x² = x⁻²，∛(x²) = x²ᐟ³。

Practice rewriting every root as an index before simplifying. This single habit prevents most errors.

在化简之前，练习把每个根式写成指数形式。这一个习惯能避免大多数错误。

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Laws of Indices | 指数法则