Simplifying Surds: From √162 to Exam Mastery | 化简根式：从√162到考试制胜

📚 Simplifying Surds: From √162 to Exam Mastery | 化简根式：从√162到考试制胜

Surds often look intimidating, but they are simply irrational numbers expressed with a root symbol. In this article, we use a single example — √162 — to unlock every key skill you need for Edexcel IGCSE Mathematics, from prime factorisation to rationalising the denominator.

根式（surds）看起来可能令人望而生畏，但它们其实就是用根号表示的无理数。在本文中，我们将以√162这个单一例子为线索，解锁Edexcel IGCSE数学中所有关键技能，从质因数分解到分母有理化。

1. What Are Surds? | 什么是根式？

A surd is an irrational number that can be expressed as a square root, cube root, or other root of a non‑perfect power. For example, √2, √3, and √162 are all surds because their values cannot be written as exact fractions or terminating decimals.

根式是指不能用有限小数或分数精确表示的无理数，通常带有根号。例如√2、√3和√162都是根式，因为它们的值既不能写成分数，也不能写成终止小数。

In the IGCSE syllabus, you are expected to simplify surds, perform operations with them, and rationalise denominators. The key is to recognise perfect square factors inside the root.

在IGCSE考纲中，你需要学会化简根式、对根式进行四则运算，并且能够将分母有理化。关键在于识别根号内的完全平方因子。

2. Prime Factorisation: The Foundation | 质因数分解：根基

To simplify a square root, first express the number as a product of its prime factors. For 162, we use a factor tree:

要化简平方根，首先把被开方数写成质因数乘积的形式。以162为例，列出因子树：

162 = 2 × 3 × 3 × 3 × 3 = 2 × 3⁴

Notice that 3⁴ is a perfect fourth power, but for square roots we look for pairs of identical factors. Since 3⁴ = (3²)² = 9², it is a perfect square.

注意3⁴是四次幂，但对于平方根我们寻找相同因子的配对。因为3⁴ = (3²)² = 9²，所以它本身是一个完全平方数。

English: Always start with prime factorisation when simplifying surds.
中文：化简根式时，始终以质因数分解作为第一步。

3. Simplifying √162 Step by Step | 逐步化简√162

Using the prime factorisation from above, we separate the perfect squares from the rest:

利用上述质因数分解，我们把完全平方因子与其他因子分开：

√162 = √(81 × 2) = √81 × √2 = 9√2

We rewrite 162 as 81 × 2 because 81 is the largest perfect square factor (81 = 9²). This leaves √2, which cannot be simplified further.

我们把162改写为81×2，因为81是最大的完全平方因子（81=9²）。剩余部分√2已不能再化简。

Always check your answer by squaring it: (9√2)² = 81 × 2 = 162. This confirms correctness.

务必通过平方来检验答案：(9√2)² = 81 × 2 = 162，这样就能确认正确性。

4. Multiplying and Dividing Surds | 根式的乘法与除法

The rules for multiplying surds are straightforward: √a × √b = √(ab), and similarly for division √a ÷ √b = √(a/b). These rules hold when a and b are positive.

根式乘除法的规则很简单：√a × √b = √(ab)，除法类似√a ÷ √b = √(a/b)。当a和b均为正数时这些规则成立。

For example, simplify √6 × √18:

例如，化简√6 × √18：

√6 × √18 = √(6 × 18) = √108 = √(36 × 3) = 6√3

For division, consider √72 ÷ √2:

对于除法，看√72 ÷ √2：

√72 ÷ √2 = √(72 ÷ 2) = √36 = 6

Always simplify the final answer by checking for perfect square factors.

最后务必检查结果中是否还有完全平方因子，并化简到最简形式。

5. Adding and Subtracting Surds | 根式的加法与减法

Surds can be added or subtracted only when they have the same irrational part. For instance, 3√2 + 5√2 = 8√2. Simplify each surd first to see if like terms appear.

只有具有相同无理部分的根式才能相加减。例如3√2 + 5√2 = 8√2。先化简每个根式，再看是否出现同类项。

Consider √162 + √32:

考察√162 + √32：

√162 = 9√2, √32 = 4√2, so 9√2 + 4√2 = 13√2

If the irrational parts differ, such as √2 + √3, the expression cannot be simplified further.

如果无理部分不同，例如√2 + √3，则这个表达式不能再化简。

English: Simplify each surd separately before adding or subtracting.
中文：在加减之前先分别化简每个根式。

6. Expanding Brackets with Surds | 用根式展开括号

You may need to expand expressions like (√5 + 2)(√5 − 1). Apply the distributive law carefully, treating surds as algebraic terms.

你可能会遇到类似(√5 + 2)(√5 − 1)的展开式。仔细运用分配律，把根式当作代数项来处理。

(√5 + 2)(√5 − 1) = √5 × √5 − √5 × 1 + 2 × √5 − 2 × 1

Simplify each product: √5 × √5 = 5, then collect like terms:

化简每一项：√5 × √5 = 5，然后合并同类项：

= 5 − √5 + 2√5 − 2 = 3 + √5

Remember that (√a)² = a. This is a fundamental property used in many expansions.

记住(√a)² = a。这是许多展开式中会用到的根本性质。

7. Rationalising the Denominator: Basic Case | 分母有理化：基础情形

It is considered neater to have a rational number in the denominator. To rationalise a fraction like 1/√2, multiply numerator and denominator by √2:

通常将分母化为有理数更整洁。要把像1/√2这样的分数有理化，需将分子分母同时乘以√2：

1/√2 × √2/√2 = √2/2

Similarly, for 5/√162, first simplify √162:

类似地，对于5/√162，先化简√162：

√162 = 9√2, so 5/√162 = 5/(9√2)

Now multiply numerator and denominator by √2:

然后分子分母同乘√2：

5/(9√2) × √2/√2 = 5√2/(9 × 2) = 5√2/18

The denominator is now rational (18 is a rational number).

此时分母已经是有理数（18是有理数）。

8. Rationalising: Conjugate Pairs | 有理化：共轭对

When the denominator is a binomial expression containing a surd, such as 1/(1 + √2), multiply by its conjugate. The conjugate of a + b√c is a − b√c.

当分母是包含根式的二项式，例如1/(1 + √2)，需要乘以它的共轭式。a + b√c的共轭式为a − b√c。

The product of a conjugate pair gives a rational number: (a + b√c)(a − b√c) = a² − b²c.

共轭对的乘积是有理数：(a + b√c)(a − b√c) = a² − b²c。

Example: Simplify 1/(1 + √2).

例子：化简1/(1 + √2)。

1/(1 + √2) × (1 − √2)/(1 − √2) = (1 − √2)/(1 − 2) = (1 − √2)/(−1) = √2 − 1

This technique is essential for solving trigonometric equations and evaluating exact values in IGCSE.

这一技巧在IGCSE的三角函数方程和精确值计算中至关重要。

9. Surds and Indices: The Hidden Link | 根式与指数：隐秘的联系

Every surd can be written as a fractional index: √a = a^(1/2), ³√a = a^(1/3). This connects surds directly to the laws of indices that you already know.

每一个根式都可以写成分数指数：√a = a^(1/2)，³√a = a^(1/3)。这就将根式与已有的指数运算律直接联系起来。

For example, (√162)³ = (9√2)³ = 729 × 2√2 = 1458√2, or using indices: (162^(1/2))³ = 162^(3/2).

例如，(√162)³ = (9√2)³ = 729 × 2√2 = 1458√2，或者用指数表示：(162^(1/2))³ = 162^(3/2)。

Be careful when combining roots and powers: always simplify the surd first if possible.

当根式与幂结合时需小心：如果可能，先化简根式再做进一步运算。

10. Common Exam Mistakes and How to Avoid Them | 常见考试错误及避免方法

Many students lose marks on surds because of simple oversights. Here are the top pitfalls:

很多学生在根式题目上失分，皆因粗心大意。以下是常见陷阱：

Mistake \| 错误	Correct Approach \| 正确做法
√(a + b) = √a + √b	√(a + b) cannot be simplified in general. No such rule.
Forgetting to simplify a surd entirely, e.g. leaving √32 as √(16×2) instead of 4√2.	Always extract the largest perfect square factor.
Adding 2√3 + 3√2 = 5√5 (incorrect).	Only like surds can be added: 2√3 + 3√3 = 5√3.
Multiplying numerator only when rationalising.	Multiply both numerator and denominator by the same factor.

Always check your final answers by squaring them or substituting back into the original expression.

务必通过平方或回代原式来检查最终答案。

11. Practice Questions: Test Your Skills | 练习题：检验你的技能

Try these questions from past Edexcel IGCSE papers. Solutions are provided below.

试试下面这些来自往年Edexcel IGCSE试卷的题目。解答在下方给出。

English: 1. Fully simplify √162.
中文：1. 化简√162。
English: 2. Simplify √98 − √50.
中文：2. 化简√98 − √50。
English: 3. Express 2/(√3 − 1) with a rational denominator.
中文：3. 将2/(√3 − 1)化为分母有理化形式。

Solutions: 1. 9√2; 2. 7√2 − 5√2 = 2√2; 3. 2(√3 + 1)/(3 − 1) = √3 + 1

Did you get them all correct? If not, review the relevant section above.

你是否全部答对了？如果没有，请回顾上面的相应章节。

12. Summary and Final Tips | 总结与致胜建议

Surds are a finite, testable topic. Master these four core skills: simplification, four operations, rationalisation, and the link to indices.

根式是一个有限且可考的主题。掌握这四项核心技能：化简、四则运算、有理化以及与指数的联系。

English: Always look for square factors — e.g. √162 = 9√2.
中文：始终寻找平方因子——例如√162 = 9√2。
English: Only add/subtract like surds.
中文：只有同类根式才能相加减。
English: To rationalise, multiply by the conjugate or the same surd.
中文：有理化时乘以共轭式或同根式。
English: Practice with past paper questions to build speed and confidence.
中文：通过历年真题练习来提升速度和信心。

With consistent practice, you can turn the seemingly messy world of surds into clear, exact arithmetic.

通过持续练习，你就能把看似混乱的根式世界变成清晰准确的算术。

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Simplifying Surds: From √162 to Exam Mastery | 化简根式：从√162到考试制胜