Simplifying Surds and Operations | 根式化简与运算技巧

📚 Simplifying Surds and Operations | 根式化简与运算技巧

Surds are irrational numbers that can be expressed under a square root sign, such as √2, √3 or √5. In IGCSE Mathematics, you need to manipulate surds fluently, especially when solving equations, working with quadratic expressions, and handling exact values in geometry.

根式（surds）是指无法化为整数的平方根无理数，例如 √2、√3、√5 等。在 IGCSE 数学中，熟练化简和运算根式非常重要，尤其是在解方程、处理二次表达式以及几何题中使用精确值时。

1. What Are Surds? | 什么是根式

A surd is an irrational root of a positive integer. For example, √4 = 2 is rational, so it is not a surd. But √2 ≈ 1.414… is irrational and is therefore a surd.

根式是指正整数开不尽方的根，属于无理数。例如 √4 = 2 是有理数，所以不是根式；而 √2 ≈ 1.414… 是无理数，因此是根式。

√2, √3, √5, √7 are surds.
√4 = 2 is not a surd.
√9 = 3 is not a surd.

√2、√3、√5、√7 都是根式。
√4 = 2 不是根式。
√9 = 3 不是根式。

2. Basic Simplification Rule | 化简的基本规则

To simplify a surd, we use the rule √(a × b) = √a × √b. This allows us to split a square root into smaller factors, especially when one factor is a perfect square.

化简根式时使用规则 √(a × b) = √a × √b，这使我们能将一个根号拆成更小的因式，尤其当其中一个因式是完全平方数时。

√(ab) = √a × √b and √(a/b) = √a / √b

For example, √50 can be written as √(25 × 2) = √25 × √2 = 5√2.

例如，√50 可以写成 √(25 × 2) = √25 × √2 = 5√2。

3. Step-by-Step Simplification | 分步化简方法

When simplifying a square root, find the largest perfect square factor of the number under the root, then take that factor out.

化简平方根时，先找出根号下数字的最大完全平方因数，然后将其提到根号外面。

Example: Simplify √72.

示例：化简 √72。

Find the largest square factor: 72 = 36 × 2.
Write √72 = √(36 × 2).
Take the square root of 36: 6√2.

找出最大平方因数：72 = 36 × 2。
写成 √72 = √(36 × 2)。
对 36 开方：6√2。

√72 = 6√2

Always check whether the number outside the root and the number inside the root share any common square factor. If not, the surd is fully simplified.

化简后检查根号外的整数与根号内的数是否还有公有的平方因数。如果没有，则根式已最简。

4. Adding and Subtracting Like Surds | 同类根式的加减

Like surds are surds with the same number inside the square root. They can be combined just like algebraic like terms.

同类根式是指根号内数字相同的根式，它们可以像代数式中的同类项一样合并。

a√k + b√k = (a + b)√k

Example: 5√3 + 2√3 = (5 + 2)√3 = 7√3.

例如：5√3 + 2√3 = (5 + 2)√3 = 7√3。

If the surds are not alike, simplify them first. For instance, √8 + √18 can be simplified to 2√2 + 3√2 = 5√2.

如果根式不是同类，先分别化简。例如 √8 + √18 可化为 2√2 + 3√2 = 5√2。

5. Multiplying Surds | 根式的乘法

To multiply two surds, multiply the numbers under the square root signs together. If there are coefficients outside the roots, multiply the coefficients separately.

根式相乘时，把根号内的数字相乘；如果根号前有系数，则系数单独相乘。

x√a × y√b = xy√(ab)

Example: 2√3 × 5√6 = (2 × 5) × √(3 × 6) = 10√18.

例如：2√3 × 5√6 = (2 × 5) × √(3 × 6) = 10√18。

Then simplify √18 = √(9 × 2) = 3√2, so 10√18 = 10 × 3√2 = 30√2.

然后化简 √18 = √(9 × 2) = 3√2，因此 10√18 = 10 × 3√2 = 30√2。

6. Dividing Surds | 根式的除法

For division, separate the square roots and simplify the fraction if possible.

根式相除时，先把根号分别相除，再化简分数。

√a / √b = √(a/b)

Example: √20 / √5 = √(20/5) = √4 = 2.

例如：√20 / √5 = √(20/5) = √4 = 2。

If the denominator contains a surd, we often need to rationalise the denominator (see the next section).

如果分母含有根式，通常需要将分母有理化（见下一节）。

7. Rationalising the Denominator: Basic Cases | 分母有理化：基本情形

Rationalising the denominator means removing the surd from the bottom of a fraction. In IGCSE, you are expected to leave answers with rational denominators.

分母有理化就是去掉分数中分母的根号。在 IGCSE 考试中，答案通常需要保持分母为有理数。

To remove a simple √k from the denominator, multiply both numerator and denominator by √k.

要去掉分母中的简单 √k，可将分子分母同时乘以 √k。

1 / √k = (1 × √k) / (√k × √k) = √k / k

Example: 1/√2 = √2/2

例如：1/√2 = √2/2。

Example with a coefficient: 3/√6 = (3√6)/6 = √6/2 (after dividing top and bottom by 3).

带系数的例子：3/√6 = (3√6)/6 = √6/2（上下同除以 3）。

8. Rationalising Using Conjugates | 使用共轭式有理化

When the denominator has the form a + √b or a − √b, we multiply top and bottom by the conjugate. The conjugate of a + √b is a − √b, and vice versa.

当分母具有 a + √b 或 a − √b 的形式时，需要乘以它的共轭式。a + √b 的共轭是 a − √b，反之亦然。

(a + √b)(a − √b) = a² − b

Example: Simplify 2 / (1 + √3).

例如：化简 2 / (1 + √3)。

Multiply by (1 − √3)/(1 − √3).
Denominator: (1 + √3)(1 − √3) = 1² − 3 = −2.
Numerator: 2(1 − √3) = 2 − 2√3.
Result: (2 − 2√3) / (−2) = −1 + √3 = √3 − 1.

乘以 (1 − √3)/(1 − √3)。
分母：(1 + √3)(1 − √3) = 1² − 3 = −2。
分子：2(1 − √3) = 2 − 2√3。
结果：(2 − 2√3) / (−2) = −1 + √3 = √3 − 1。

9. Expanding Square Binomials with Surds | 含根式的平方展开

You may be asked to expand expressions like (√a + √b)² or (2 + √3)². Use either the distributive law or the binomial square formula.

题目可能要求展开形如 (√a + √b)² 或 (2 + √3)² 的式子。可以使用乘法分配律或完全平方公式。

(√a + √b)² = a + b + 2√(ab)

Example: (2 + √3)² = 2² + 2 × 2 × √3 + (√3)² = 4 + 4√3 + 3 = 7 + 4√3.

例如：(2 + √3)² = 2² + 2 × 2 × √3 + (√3)² = 4 + 4√3 + 3 = 7 + 4√3。

Notice that (√3)² = 3, which is a rational number.

注意 (√3)² = 3，这是有理数。

10. Expanding Products of Conjugates | 共轭乘积的展开

Conjugate products are especially useful because they produce rational numbers when the surd parts cancel.

共轭式相乘特别有用，因为根号部分会抵消，得到有理数。

(√a + √b)(√a − √b) = a − b

Example: (√5 + √2)(√5 − √2) = (√5)² − (√2)² = 5 − 2 = 3.

例如：(√5 + √2)(√5 − √2) = (√5)² − (√2)² = 5 − 2 = 3。

This technique is also the quickest way to rationalise a denominator of the form (√a + √b).

这个技巧也是将形如 (√a + √b) 的分母有理化的最快方法。

11. Solving Simple Surd Equations | 解简单根式方程

Equations containing a single surd can be solved by isolating the square root and then squaring both sides. Always check your answers because squaring can introduce extra solutions.

含单个根式的方程可以通过先移项，使根号单独在一边，然后两边平方来求解。注意平方可能引入增根，因此必须检验答案。

Example: Solve √(x + 3) = 5.

示例：解方程 √(x + 3) = 5。

Square both sides: x + 3 = 25.
Solve for x: x = 22.
Check: √(22 + 3) = √25 = 5, so x = 22 is valid.

两边平方：x + 3 = 25。
解得：x = 22。
检验：√(22 + 3) = √25 = 5，所以 x = 22 是有效解。

If the equation has two surds, square twice after isolating one surd at a time.

如果方程含两个根式，需要先移项使一个根式单独在一边，平方后再处理另一个。

12. Common Mistakes and Exam Tips | 常见错误与考试技巧

The most common mistake is believing that √(a + b) = √a + √b. This is false. For example, √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7.

最常见的错误是认为 √(a + b) = √a + √b 成立。这是错误的。例如 √(9 + 16) = √25 = 5，而 √9 + √16 = 3 + 4 = 7。

Always simplify surds fully before combining like terms.
When squaring equations, remember to check for extraneous roots.
Learn the small square numbers: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
If a calculator is not allowed, rationalise denominators to make comparisons easier.

在合并同类根式前，务必先将每个根式化简到最简。
方程两边平方后，一定要检验是否产生增根。
记住常见完全平方数：4、9、16、25、36、49、64、81、100、121、144。
如果考试不允许使用计算器，有理化分母会使后续比较或计算更简单。

Practice converting between forms such as 3√2 and √18. In many exam questions, the same value can be written in different ways, and recognising this helps you simplify complicated expressions.

多练习在 3√2 与 √18 等不同形式之间转换。在许多考题中，同一个数值可以有多种写法，识别这些等价形式能帮助你化简复杂表达式。

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Simplifying Surds and Operations | 根式化简与运算技巧