📚 Surds | 根式与无理数
Surds are exact irrational roots that appear throughout A-Level Maths, from simplifying algebraic expressions to trigonometry and coordinate geometry. This revision guide covers the Edexcel AS/A-Level techniques for manipulating surds, rationalising denominators, and answering exam questions with confidence.
根式是贯穿 A-Level 数学的精确无理根，从化简代数式到三角学和坐标几何都会出现。本复习指南涵盖 Edexcel AS/A-Level 中处理根式、分母有理化以及自信应对考试题的技巧。
1. What Are Surds? | 什么是根式
In A-Level Maths, a surd is an irrational root of a rational number. The most common surds are square roots such as √2, √3, √5, and √10, which cannot be written as exact terminating or recurring decimals. Writing √2 is exact; writing 1.41421356 is only an approximation.
在 A-Level 数学中，根式是有理数的无理根。最常见的根式是平方根，例如 √2、√3、√5 和 √10，它们都无法写成精确的有限小数或循环小数。写 √2 是精确值；写 1.41421356 只是近似值。
Numbers like √4 = 2 and √9/4 lify to whole numbers or fractions. Exam questions often expect you to leave answers in surd form so that the value remains exact.
像 √4 = 2 和 √9/4 = 3/2 这样的数不是根式，因为它们可以化简为整数或分数。考试常见要求是将答案保留为根式形式，以保证数值精确。
2. The Golden Rules of Surds | 根式运算的黄金法则
For any positive numbers a and b, the following rules always apply:
对于任意正数 a 和 b，以下法则始终成立：
- √ab = √a√b
- √a/b = √a / √b
- (√a)² = a
These rules allow you to simplify, multiply, and divide surds. However, the most dangerous mistake is assuming that √a + √b equals √a + b. That is false, and Edexcel exams frequently test this misconception.
这些法则允许你对根式进行化简、乘法和除法。然而，最危险的错误是假设 √a + √b 等于 √a + b。这是错误的，也是 Edexcel 考试经常考查的误区。
√a + √b ≠ √a + b
√a - √b ≠ √a - b
Always simplify each surd first before adding or subtracting, because only like surds can be combined.
在加减之前一定要先化简每个根式，因为只有同类根式才能合并。
3. Simplifying Surds by Factorising | 通过因式分解化简根式
The key skill is to factor out the largest square number from under the square root. For example, to simplify √72, write 72 as 36 × 2, because 36 is a perfect square.
关键技巧是从平方根下分解出最大的平方数。例如，要化简 √72，将 72 写成 36 × 2，因为 36 是完全平方数。
√72 = √36 × 2 = √36 × √2 = 6√2
Useful square numbers to recognise are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. Always check whether the radicand has a square factor larger than 1 before you stop simplifying.
需要识别的平方数有 1、4、9、16、25、36、49、64、81 和 100。在停止化简前，请始终检查被开方数是否有大于 1 的平方因数。
Another example is √48 = √16 × 3 = 4√3. If you chose √4 × 12, you would get 2√12, but that is not fully simplified because 12 still contains the square factor 4.
另一个例子是 √48 = √16 × 3 = 4√3。如果你选择 √4 × 12，会得到 2√12，但这并没有完全化简，因为 12 还含有平方因数 4。
4. Adding and Subtracting Surds | 根式的加减法
Only like surds can be added or subtracted. Like surds have the same number under the square root. For example, 3√2 and 5√2 are like surds, so they can be combined easily.
只有同类根式才能相加或相减。同类根式的平方根下数字相同。例如，3√2 和 5√2 是同类根式，因此可以直接合并。
3√2 + 5√2 = 8√2
If the surds look different at first, simplify them before deciding whether they are like. For instance, √8 and √18 both simplify to multiples of √2.
如果根式一开始看起来不同，请先化简再判断它们是否为同类。例如，√8 和 √18 化简后都变成 √2 的倍数。
√8 + √18 = 2√2 + 3√2 = 5√2
You cannot combine √2 and √3 into a single surd. Leave the answer as √2 + √3, or write it in its simplest separated form.
你不能把 √2 和 √3 合并成一个根式。答案应保留为 √2 + √3，或写成最简的分开形式。
5. Multiplying and Dividing Surds | 根式的乘除法
Multiplication and division follow the golden rules directly. When multiplying two single surds, multiply the numbers inside the square roots and simplify if possible.
乘法和除法直接遵循黄金法则。两个单个根式相乘时，将平方根内的数相乘，然后尽可能化简。
√3 × √5 = √3 × 5 = √15
For surds with coefficients, multiply the coefficients together and the radicands together separately.
对于带系数的根式，将系数与系数相乘、被开方数与被开方数相乘。
(2√3) × (5√2) = 2 × 5 × √3 × 2 = 10√6
Division works in a similar way. Divide the coefficients and simplify the fraction inside the square root.
除法的处理方式类似。将系数相除，并化简平方根内的分数。
√18 ÷ √2 = √18 ÷ 2 = √9 = 3
6√10 ÷ 2√5 = 3√10 ÷ 5 = 3√2
Always check whether the result can be simplified further by looking for square factors.
始终检查结果是否还能通过寻找平方因数进一步化简。
6. Expanding Brackets with Surds | 含根式括号的展开
Expanding brackets with surds uses the same FOIL method as ordinary algebra. For example, expand (√2 + 3)(√2 - 1).
含根式括号的展开与普通代数一样使用 FOIL 方法。例如，展开 (√2 + 3)(√2 - 1)。
(√2 + 3)(√2 - 1) = (√2)² - √2 + 3√2 - 3 = 2 + 2√2 - 3 = -1 + 2√2
The difference of two squares is extremely useful because it removes square roots from products such as (√a + √b)(√a - √b).
平方差公式极其有用，因为它能从 (√a + √b)(√a - √b) 这类乘积中消去平方根。
(√a + √b)(√a - √b) = a - b
For example, (√5 + √3)(√5 - √3) = 5 - 3 = 2. This idea is central to rationalising denominators in later topics.
例如，(√5 + √3)(√5 - √3) = 5 - 3 = 2。这一思想是后面分母有理化的核心。
When squaring a sum of surds, use (a + b)² = a² + 2ab + b². Do not forget the middle term.
当对根式之和进行平方时，使用 (a + b)² = a² + 2ab + b²。不要漏掉中间项。
(√3 + √5)² = 3 + 2√15 + 5 = 8 + 2√15
7. Rationalising a Monomial Denominator | 单项式分母有理化
A fraction is not considered fully simplified if its denominator contains a surd. Rationalising the denominator removes the surd by multiplying both the numerator and denominator by the same root.
如果分数的分母含有根式，则它不被视为完全化简。分母有理化通过将分子和分母同时乘以相同的根式来消去根式。
1 / √2 = 1/√2 × √2/√2 = √2 / 2
For a denominator of the form a√b, multiply the top and bottom by √b. This works because √b × √b = b.
对于形如 a√b 的分母，将分子和分母同时乘以 √b。这是可行的，因为 √b × √b = b。
5 / (2√3) = 5/(2√3) × √3/√3 = 5√3 / (2 × 3) = 5√3 / 6
Always simplify the final fraction if the numerator and denominator share common factors.
如果最终分数的分子和分母有公因数，一定要进行约分。
8. Rationalising a Binomial Denominator | 二项式分母有理化
When the denominator is a sum or difference of two terms, such as √3 - 1, you must multiply by its conjugate. The conjugate of √3 - 1 is √3 + 1, and their product removes the surds.
当分母是两项的和或差（例如 √3 - 1）时，你必须乘以其共轭式。√3 - 1 的共轭式是 √3 + 1，它们的乘积可以消去根式。
1 / (√3 - 1) = 1/(√3 - 1) × (√3 + 1)/(√3 + 1) = (√3 + 1) / (3 - 1) = (√3 + 1) / 2
This uses the difference of two squares: (√3 - 1)(√3 + 1) = (√3)² - 1² = 3 - 1 = 2. The method works for any binomial denominator such as a ± √b or √a ± √b.
这里使用平方差公式：(√3 - 1)(√3 + 1) = (√3)² - 1² = 3 - 1 = 2。该方法适用于任何二项式分母，例如 a ± √b 或 √a ± √b。
A more complex example is (√5 + √2) / (√5 - √2). Multiply the numerator and denominator by √5 + √2.
一个更复杂的例子是 (√5 + √2) / (√5 - √2)。将分子和分母同时乘以 √5 + √2。
(√5 + √2) / (√5 - √2) = (√5 + √2)² / (5 - 2) = (5 + 2√10 + 2) / 3 = (7 + 2√10) / 3
Never multiply a binomial denominator by just one of its terms; you must use the full conjugate to eliminate the root completely.
切勿只将二项式分母乘以其某一项；必须使用完整的共轭式才能彻底消去根号。
9. Solving Equations with Surds | 解含根式的方程
To solve an equation such as √x + 1 = 3, square both sides and then solve the resulting linear or quadratic equation.
要求解像 √x + 1 = 3 这样的方程，先将两边平方，然后解得到的线性或二次方程。
√x + 1 = 3
x + 1 = 9
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图