IGCSE Maths: Indices (Powers & Roots) | 指数与根式精讲

📚 IGCSE Maths: Indices (Powers & Roots) | 指数与根式精讲

Indices, also known as powers or exponents, are a fundamental concept in IGCSE Mathematics. They appear in algebra, geometry, and even statistics. Mastering indices is essential for solving equations, simplifying expressions, and understanding more advanced topics such as exponential growth and logarithms.

指数（Indices），也称为幂或指数项，是 IGCSE 数学中的一个基础概念。它们出现在代数、几何甚至统计学中。掌握指数对于解方程、化简表达式以及理解指数增长和对数等高阶主题至关重要。

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1. What are Indices? | 什么是指数？

An index (plural: indices) tells us how many times a number (the base) is multiplied by itself. For example, in the expression 2³, the base is 2 and the index is 3. This means 2 × 2 × 2 = 8.

指数告诉我们一个数（底数）自乘多少次。例如，在表达式 2³ 中，底数是 2，指数是 3。这意味着 2 × 2 × 2 = 8。

The general form is written as aⁿ, where ‘a’ is the base and ‘n’ is the index. In IGCSE exams, you will encounter questions that test your ability to evaluate, simplify, and solve expressions involving indices.

一般形式写作 aⁿ，其中 a 是底数，n 是指数。在 IGCSE 考试中，你会遇到测试你计算、化简和求解指数表达式能力的题目。

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2. The Basic Laws of Indices | 基本指数法则

There are several key laws of indices that you must know. These rules allow you to simplify expressions involving powers of the same base.

有几个必须掌握的关键指数法则。这些规则允许你化简涉及同底数幂的表达式。

Law 1: Multiplication When multiplying two powers with the same base, add the indices.

法则 1：乘法当两个同底数的幂相乘时，指数相加。

aᵐ × aⁿ = aᵐ⁺ⁿ

For example: 2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128.

例如：2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128。

Law 2: Division When dividing two powers with the same base, subtract the indices.

法则 2：除法当两个同底数的幂相除时，指数相减。

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

For example: 5⁶ ÷ 5² = 5⁶⁻² = 5⁴ = 625.

例如：5⁶ ÷ 5² = 5⁶⁻² = 5⁴ = 625。

Law 3: Power of a Power When raising a power to another power, multiply the indices.

法则 3：幂的幂当一个幂再被乘方时，指数相乘。

(aᵐ)ⁿ = aᵐⁿ

For example: (3²)⁴ = 3²ˣ⁴ = 3⁸ = 6561.

例如：(3²)⁴ = 3²ˣ⁴ = 3⁸ = 6561。

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3. More Laws of Indices | 更多指数法则

Beyond the first three, there are additional laws that are equally important.

除了前三条法则，还有其他同样重要的法则。

Law 4: Power of a Product The power of a product is the product of the powers.

法则 4：积的幂积的幂等于各个因子的幂的乘积。

(ab)ⁿ = aⁿbⁿ

For example: (2 × 3)² = 2² × 3² = 4 × 9 = 36.

例如：(2 × 3)² = 2² × 3² = 4 × 9 = 36。

Law 5: Power of a Quotient The power of a quotient is the quotient of the powers.

法则 5：商的幂商的幂等于分子分母各自幂的商。

(a/b)ⁿ = aⁿ / bⁿ

For example: (4/5)² = 4² / 5² = 16/25.

例如：(4/5)² = 4² / 5² = 16/25。

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4. Zero and Negative Indices | 零指数与负指数

Two special cases are the zero index and negative indices.

两个特殊的情况是零指数和负指数。

Zero Index: Any non-zero number raised to the power of 0 is equal to 1.

零指数：任何非零数的零次方都等于 1。

a⁰ = 1 (where a ≠ 0)

For example: 7⁰ = 1, (x/y)⁰ = 1.

例如：7⁰ = 1，(x/y)⁰ = 1。

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

负指数：负指数表示底数正指数幂的倒数。

a⁻ⁿ = 1 / aⁿ

For example: 2⁻³ = 1/2³ = 1/8.

例如：2⁻³ = 1/2³ = 1/8。

A common mistake is thinking that a negative index makes the answer negative. This is not the case.

一个常见错误是认为负指数会让答案变成负数。事实并非如此。

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5. Fractional Indices | 分数指数

Fractional indices are another essential topic. They are linked directly to roots.

分数指数是另一个重要主题。它们与根式直接相关。

The 1/n power: The index 1/n represents the nth root of the base.

1/n 次幂：指数 1/n 表示底数的 n 次方根。

a¹/ⁿ = ⁿ√a

For example: 9¹/² = √9 = 3, and 27¹/³ = ∛27 = 3.

例如：9¹/² = √9 = 3，27¹/³ = ∛27 = 3。

The m/n power: The index m/n can be interpreted in two ways: (ⁿ√a)ᵐ or ⁿ√(aᵐ). Both give the same answer.

m/n 次幂：指数 m/n 有两种理解方式：(ⁿ√a)ᵐ 或 ⁿ√(aᵐ)。两者结果相同。

aᵐ/ⁿ = (ⁿ√a)ᵐ = ⁿ√(aᵐ)

For example: 8²/³ = (∛8)² = 2² = 4. It is often easier to take the root first.

例如：8²/³ = (∛8)² = 2² = 4。通常先开方再乘方会更简单。

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6. Simplifying Expressions | 化简表达式

In exams, you will be asked to simplify algebraic expressions that contain indices. The key is to apply the laws systematically and work step by step.

在考试中，你会被要求化简含有指数的代数表达式。关键在于系统性地应用法则并逐步计算。

Consider this example: Simplify (2x²y³)².

考虑这个例子：化简 (2x²y³)²。

Apply the power to each factor inside the bracket:

将幂作用到括号内的每个因子：

(2x²y³)² = 2² × (x²)² × (y³)² = 4x⁴y⁶

Another example: Simplify (3a⁵b²) ÷ (a²b⁷).

另一个例子：化简 (3a⁵b²) ÷ (a²b⁷)。

Divide the coefficients and subtract the indices of like bases:

系数相除，同底数的指数相减：

3a⁵⁻²b²⁻⁷ = 3a³b⁻⁵ = 3a³ / b⁵

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7. Solving Equations with Indices | 解指数方程

When the unknown appears as an index, we can often solve the equation by making the bases the same.

当未知数出现在指数位置时，我们通常可以通过使底数相同来解方程。

Example: Solve 2ˣ = 32.

例子：解方程 2ˣ = 32。

Write 32 as a power of 2: 32 = 2⁵. Therefore x = 5.

将 32 写成 2 的幂：32 = 2⁵。因此 x = 5。

Example with fractions: Solve 3ˣ = 1/81.

含分数的例子：解方程 3ˣ = 1/81。

Since 1/81 = 1/3⁴ = 3⁻⁴, we have x = -4.

因为 1/81 = 1/3⁴ = 3⁻⁴，所以 x = -4。

For equations like 4ˣ = 8, write both sides with the same base:

对于类似 4ˣ = 8 的方程，将两边写成相同的底数：

4ˣ = (2²)ˣ = 2²ˣ, and 8 = 2³, so 2²ˣ = 2³, hence x = 1.5

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8. Common Exam Question Types | 常见考试题型

Let’s look at some typical IGCSE exam questions

让我们来看一些典型的 IGCSE 考试题目。

Question 1: Evaluate (25)³/².

题目 1：计算 (25)³/²。

Solution: (25)³/² = (√25)³ = 5³ = 125.

解答：(25)³/² = (√25)³ = 5³ = 125。

Question 2: Simplify (x²y⁴)³ / x⁵y².

题目 2：化简 (x²y⁴)³ / x⁵y²。

Solution: (x²y⁴)³ = x⁶y¹², so x⁶⁻⁵y¹²⁻² = xy¹⁰.

解答：(x²y⁴)³ = x⁶y¹²，所以 x⁶⁻⁵y¹²⁻² = xy¹⁰。

Question 3: Solve 5²ˣ = 125.

题目 3：解方程 5²ˣ = 125。

Solution: 125 = 5³, so 2x = 3, x = 1.5.

解答：125 = 5³，因此 2x = 3，x = 1.5。

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9. Practice Questions | 模拟练习

Test yourself with these practice questions. Attempt them before looking at the answers.

用以下练习题测试自己。先尝试作答，再看答案。

1. Evaluate 64¹/².

1. 计算 64¹/²。

2. Evaluate 16⁻¹/².

2. 计算 16⁻¹/²。

3. Simplify (4a³)² ÷ (2a)².

3. 化简 (4a³)² ÷ (2a)²。

4. Express ⁴√x³ as a power of x.

4. 将 ⁴√x³ 表示为 x 的幂。

5. Solve 2ˣ = 64.

5. 解方程 2ˣ = 64。

6. Solve 9ˣ = 3.

6. 解方程 9ˣ = 3。

Answers: 1. 8 2. 1/4 3. 4a⁴ 4. x³/⁴ 5. x = 6 6. x = 0.5

答案：1. 8 2. 1/4 3. 4a⁴ 4. x³/⁴ 5. x = 6 6. x = 0.5

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10. Summary of Key Rules | 关键规则总结

Here is a quick reference table for all the laws of indices you need for IGCSE.

以下是你 IGCSE 需要的所有指数法则的快速参考表。

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