Example 1.2.1: Simplifying Surds | 例题1.2.1：化简根式

📚 Example 1.2.1: Simplifying Surds | 例题1.2.1：化简根式

In this revision article, we break down a classic AQA A-Level Maths example from Chapter 1, Section 2. Example 1.2.1 focuses on simplifying surds, a core skill for algebraic manipulation in the AS and A-Level examinations.

本复习文章将详细拆解 AQA 数学 A-Level 第1章第2节中的一道经典例题。例题1.2.1 重点训练根式化简，这是 AS 及 A-Level 考试中代数运算的核心技能。

1. What Is a Surd? | 什么是根式

A surd is an irrational number that can be expressed as a root of a positive integer, such as √2, √3 or √5. These numbers cannot be written as exact terminating or recurring decimals.

根式是一种无理数，可表示为正整数开根的形式，例如 √2、√3 或 √5。这些数无法写成精确的有限小数或循环小数。

Simplifying surds means rewriting them in a simpler, equivalent form. The goal is to express the surd so that the number inside the root has no square factors greater than 1.

化简根式意味着将其改写为更简单的等价形式。目标是使得根号内的数不再含有大于1的平方因子。

2. The Example Statement | 例题描述

In Example 1.2.1, you are typically asked to simplify the following expression:

例题1.2.1 通常要求化简以下表达式：

√48 − √12 + √27

This is a direct test of prime factorisation and the ability to combine like surds. We will solve it step by step.

此题直接考查质因数分解以及合并同类根式的能力。我们将逐步求解。

3. Step 1: Prime Factorisation | 第1步：质因数分解

The first step is to express each number under the square root as a product of its prime factors.

第一步是将每个根号下的数表示为质因数的乘积。

48 = 2 × 2 × 2 × 2 × 3 = 16 × 3

48 = 2 × 2 × 2 × 2 × 3 = 16 × 3
12 = 2 × 2 × 3 = 4 × 3

12 = 2 × 2 × 3 = 4 × 3
27 = 3 × 3 × 3 = 9 × 3

27 = 3 × 3 × 3 = 9 × 3

We are looking for perfect squares among the factors: 16, 4 and 9 are perfect squares.

我们在因子中寻找完全平方数：16、4 和 9 都是完全平方数。

4. Step 2: Simplify Each Surd | 第2步：化简每个根式

Use the rule that √(a × b) = √a × √b to separate the perfect square from the remaining factor.

利用法则 √(a × b) = √a × √b，将完全平方因子与剩余因子分离。

√48 = √(16 × 3) = √16 × √3 = 4√3

√12 = √(4 × 3) = √4 × √3 = 2√3

√27 = √(9 × 3) = √9 × √3 = 3√3

Each original surd is now written as a number (the coefficient) times the common surd √3.

现在每个原始根式都写成了一个数字（系数）乘以共同根式 √3 的形式。

5. Step 3: Combine Like Terms | 第3步：合并同类项

Since all three simplified surds share the same irrational part √3, we can add or subtract them like terms.

由于三个化简后的根式都有相同的无理部分 √3，我们像合并同类项一样进行加减运算。

4√3 − 2√3 + 3√3 = (4 − 2 + 3)√3 = 5√3

Thus, the simplified form of √48 − √12 + √27 is exactly 5√3.

因此，√48 − √12 + √27 的化简结果恰好是 5√3。

6. Checking Your Answer | 检查答案

It is always good practice to verify your result using a calculator. Since √3 ≈ 1.732, we have:

始终建议用计算器验证结果。因为 √3 ≈ 1.732，所以：

5√3 ≈ 5 × 1.732 = 8.660

Original: √48 ≈ 6.928, √12 ≈ 3.464, √27 ≈ 5.196. Sum: 6.928 − 3.464 + 5.196 = 8.660.

原始数值：√48 ≈ 6.928，√12 ≈ 3.464，√27 ≈ 5.196。求和：6.928 − 3.464 + 5.196 = 8.660。

The two values match, confirming our simplification is correct.

两者数值一致，验证了我们的化简过程是正确的。

7. Common Mistakes | 常见错误

Students often make errors in this type of question. Here are the most common pitfalls:

学生在处理这类问题时经常犯错。以下是最常见的陷阱：

Treating √(a + b) as √a + √b. This is wrong: √(16 + 9) ≠ √16 + √9.

把 √(a + b) 错误地当作 √a + √b。这是不对的：√(16 + 9) ≠ √16 + √9。
Forgetting to simplify one of the surds completely. For example, leaving √48 as 4√3 is fine, but writing √48 = 2√12 is not fully simplified.

忘记完全化简某一个根式。例如，把 √48 写成 4√3 是正确的，但写成 2√12 则没有完全化简。
Adding coefficients incorrectly when combining terms. Keep the surd factor unchanged.

合并项时错误地相加系数。要保持根式因子不变。

8. Extension: Rationalising a Denominator | 拓展：分母有理化

A related skill is rationalising the denominator when a surd appears in the bottom of a fraction. For example:

一个相关技能是当分数分母中含有根式时进行有理化。例如：

Simplify: 3 / √2

Multiply numerator and denominator by √2:

将分子与分母同时乘以 √2：

(3 × √2) / (√2 × √2) = 3√2 / 2

This eliminates the surd on the denominator, making the expression easier to work with.

这消除了分母中的根式，使表达式更易于继续运算。

9. Exam Tips for AQA A-Level Maths | AQA 数学 A-Level 考试技巧

When encountering surds in an exam, always look for perfect square factors first. In the exam, marks are awarded for both the simplification step and the final correct answer.

在考试中遇到根式时，首先要寻找完全平方因子。在考试中，化简步骤和最终正确答案都会予给分数。

Write every intermediate step clearly. A correct method with a small arithmetic slip may still gain most of the marks.

清晰写出每一个中间步骤。若方法正确而仅仅计算笔误，通常仍能获得大部分分数。

If a question asks for an answer in the form a√b, make sure that b is a positive integer with no square factors, and a is an integer.

如果题目要求以 a√b 的形式作答，请确保 b 是一个不含平方因子的正整数，且 a 为整数。

10. Practice Problems | 练习题目

Try these problems to build confidence. Simplify each expression and then expand.

尝试以下题目来增强信心。化简每个表达式并展开。

1. √50 − √18

1. √50 − √18
2. √28 + √63 − √7

2. √28 + √63 − √7
3. (√5 + √2)(√5 − √2)

3. (√5 + √2)(√5 − √2)

Answers: 1. 2√2; 2. 4√7; 3. 3 (using difference of squares).

答案：1. 2√2；2. 4√7；3. 3（利用平方差公式）。

11. Conclusion | 总结

Example 1.2.1 demonstrates a systematic method for simplifying surds: factorise, extract perfect squares, then combine like terms. Mastery of this technique is essential for AQA A-Level Maths.

例题1.2.1 展示了化简根式的系统方法：分解因数、提取完全平方、合并同类项。掌握这一技巧对于 AQA 数学 A-Level 极为重要。

Keep practising with a range of surd expressions, and always check your final answer either by reverse calculation or approximate numerical comparison.

坚持练习各种根式表达式，并通过逆运算或近似数值比较来检查最终答案。

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Example 1.2.1: Simplifying Surds | 例题1.2.1：化简根式