📚 Rationalising Denominators | 分母有理化
Rationalising the denominator is a fundamental algebraic technique used to eliminate surds (irrational roots) from the denominator of a fraction. In A-Level Mathematics, mastering this skill is essential for simplifying expressions, solving equations, and obtaining exact values rather than decimal approximations. This article will guide you through the concept, methods, and common pitfalls in rationalising denominators, with worked examples aligned to Edexcel specifications.
分母有理化是一种重要的代数技巧，用于将分母中的根式（无理根）消去，使分母变为有理数。在A-Level数学中，掌握这一技巧对化简表达式、求解方程以及获得精确值而非近似小数至关重要。本文将详细讲解分母有理化的概念、方法和常见错误，并通过符合Edexcel考纲的例题帮助你深入理解。
1. What is Rationalising the Denominator? | 什么是分母有理化？
Rationalising the denominator means rewriting a fraction so that the denominator no longer contains any square roots (or higher-order roots). For example, the fraction 1/√2 can be rationalised to √2/2, which is an equivalent form without a surd in the denominator. This process relies on the property that multiplying a surd by itself yields a rational number.
分母有理化是指将分数改写，使分母不再含有平方根（或更高次根式）。例如，分数 1/√2 可化为 √2/2，另一种等价形式分母中不含根号。这一过程依赖于根式自乘得到有理数的性质。
In the Edexcel A-Level syllabus, rationalisation often appears in topics such as surds, coordinate geometry, trigonometry, and integration, where exact answers are required. The ability to rationalise denominators simplifies numerical expressions and prepares you for more advanced algebraic manipulations.
在Edexcel A-Level考纲中，有理化常常出现在根式、坐标几何、三角学和积分等专题中，这些地方要求给出精确答案。分母有理化的能力可以简化数值表达式，为你进行更复杂的代数运算打下基础。
2. Why Do We Rationalise Denominators? | 为什么要进行分母有理化？
Rationalising denominators serves several purposes. First, it standardises the form of an expression, making it easier to compare, add, or subtract fractions. Second, it often reveals a simpler exact value that is more convenient for further computation. Third, many examining bodies, including Edexcel, expect final answers to be given with rationalised denominators unless otherwise stated.
进行分母有理化有多个目的。首先，它使表达式形式统一，便于比较、相加或相减。其次，它通常能揭示更简单的精确值，便于后续计算。第三，包括Edexcel在内的许多考试局，除非特别说明，都要求最终答案采用分母有理化的形式。
Historically, before calculators were ubiquitous, rationalising made it easier to obtain numerical approximations by hand, because dividing by a rational number is more straightforward than dividing by an irrational one. This legacy persists in modern examinations.
在计算器普及之前，分母有理化能使人工求近似值更加容易，因为除以有理数比除以无理数简单。这一传统仍在现代考试中延续。
3. Rationalising Denominators with a Single Surd | 单一根式分母的有理化
When the denominator is a single surd such as √a, rationalisation is straightforward: multiply both the numerator and the denominator by that same surd. This works because √a × √a s rational. The fraction retains its value because you are multiplying by 1 in the form of √a/√a.
当分母为单一根式如 √a 时，有理化非常简单：将分子和分母同时乘以该根式即可。这是因为 √a × √a = a，是有理数。分数的值保持不变，因为你乘的是 √a/√a，相当于1。
General rule: a/(b√c) = a√c/(bc) after rationalisation. Always simplify any resulting surds in the numerator if possible.
一般规则：a/(b√c) 有理化后得到 a√c/(bc)。如果分子出现根式，还需进一步化简。
4. Step-by-Step Method for √a in the Denominator | 分母含 √a 的分步方法
Let's illustrate with the fraction 5/√3. Step 1: Identify the surd in the denominator (√3). Step 2: Multiply numerator and denominator by √3. This gives (5 × √3)/(√3 × √3) = 5√3/3. Step 3: Check if the fraction can be simplified further; here it cannot. The rationalised form is 5√3/3.
我们以分数 5/√3 为例。步骤1：确定分母中的根式 (√3)。步骤2：分子分母同乘 √3。得到 (5 × √3)/(√3 × √3) = 5√3/3。步骤3：检查是否可以进一步约分，此处不能。有理化后的形式为 5√3/3。
If the denominator has a coefficient multiplied by a surd, e.g., 2√5, the process is the same: multiply by √5. The coefficient remains as a factor.
如果分母是系数乘以根式，例如 2√5，方法相同：乘以 √5，系数作为因子保留。
5. Worked Example: 1/√3 and Variations | 例题：1/√3 及其变形
Example 1: Rationalise 1/√3. Multiply top and bottom by √3: (1×√3)/3 = √3/3. This is the exact form. If you needed to approximate with √3≈1.732, dividing by 3 is simpler than dividing by 1.732 directly.
例题1：将 1/√3 有理化。上下同乘 √3：(1×√3)/3 = √3/3。这是精确形式。若要用 √3≈1.732 求近似值，除以3比直接除以1.732更简便。
Example 2: Rationalise 3/(2√7). Multiply by √7: (3√7)/(2×7) = 3√7/14. Simplify: 3√7/14. No further reduction possible.
例题2：将 3/(2√7) 有理化。乘以 √7：(3√7)/(2×7) = 3√7/14。无法再约分。
Example 3: Rationalise √8/√2. First simplify if possible: √8 = 2√2. Then fraction becomes (2√2)/√2 = 2. This demonstrates that rationalising can sometimes eliminate the surd entirely.
例题3：有理化 √8/√2。先化简：√8 = 2√2，则分数变为 (2√2)/√2 = 2。这说明有理化有时可彻底消去根式。
6. Rationalising Denominators with Two Terms (Binomial Denominators) | 二项根式分母的有理化
When the denominator contains two terms involving surds, such as a + √b or √a + √b Instead, we use the difference of squares: (p + q)(p - q) = p² - q². The conjugate of (a + √b) is (a - √b), and vice versa.
当分母含有两项根式时，如 a + √b 或 √a + √b，仅仅乘以该根式无法消除无理部分。此时我们利用平方差公式：(p + q)(p - q) = p² - q²。 (a + √b) 的共轭式是 (a - √b)，反之亦然。
Multiplying the numerator and denominator by the conjugate converts the denominator into a rational number. This technique is central to handling surd expressions in algebra.
将分子和分母同时乘以共轭式，能把分母转化为有理数。这一技巧是处理代数中根式表达式的核心。
7. The Concept of Conjugates | 共轭的概念
The conjugate of a binomial expression changes the sign between the terms. For instance: - Conjugate of (3 + √2) is (3 - √2). - Conjugate of (√5 - 2) is (√5 + 2). - Conjugate of (√a + √b) is (√a - √b).
二项式的共轭式改变两项之间的符号。例如： - (3 + √2) 的共轭是 (3 - √2)。 - (√5 - 2) 的共轭是 (√5 + 2)。 - (√a + √b) 的共轭是 (√a - √b)。
These pairs are designed so that their product has no surd terms: (a + √b)(a - √b) = a² - b, and (√a + √b)(√a - √b) = a - b. In both cases, the result is rational (provided a and b are rational).
这些配对旨在确保它们的乘积不含根式：(a + √b)(a - √b) = a² - b，以及 (√a + √b)(√a - √b) = a - b。两种情况下，结果均为有理数（假设a和b是有理数）。
8. Step-by-Step Method for a + b√c Type Denominator | a + b√c 型分母的分步方法
Consider the fraction 1/(2 + √3). Step 1: Identify the denominator's conjugate (2 - √3). Step 2: Multiply numerator and denominator by this conjugate: 1×(2 - √3)/( (2+√3)(2-√3) ). Step 3: Expand the denominator using difference of squares: 2² - (√3)² = 4 - 3 = 1. The denominator becomes 1, so the fraction simplifies to (2 - √3).
考虑分数 1/(2 + √3)。步骤1：确定分母的共轭式 (2 - √3)。步骤2：分子分母同乘此共轭式：1×(2 - √3)/( (2+√3)(2
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