Rationalising Denominators | 有理化分母

📚 Rationalising Denominators | 有理化分母

Rationalising the denominator is a core surds skill in the Edexcel A-Level Mathematics specification. It is tested in Pure Mathematics, often within simplification, algebraic manipulation, and exact-value problems. This revision guide explains the method step by step, from simple square-root denominators to binomial denominators with two terms.

有理化分母是爱德思 A-Level 数学考试中根式部分的核心技能。它常见于纯数学的化简、代数变形和精确值问题中。本复习指南将逐步讲解从单一平方根分母到二项式分母的有理化方法。

1. What Is Rationalising the Denominator? | 什么是有理化分母？

Rationalising the denominator means rewriting a fraction so that the denominator is a rational number or rational expression, removing any surds from the bottom of the fraction. A surd is an irrational root, such as √2 or √5.

有理化分母是指将一个分数改写，使分母成为有理数或有理式，从而去除分数底部的根号。根式是指无理根，例如 √2 或 √5。

For example, the fraction 1/√2 has a surd in the denominator. It can be rewritten as √2/2, which has a rational denominator of 2.

例如，分数 1/√2 的分母中含有根式。它可以改写为 √2/2，此时分母 2 是有理数。

2. Why Rationalise? | 为什么要进行有理化？

In A-Level Edexcel exams, final answers are normally expected in their simplest exact form. A rationalised denominator is considered standard form and makes it easier to compare, add, subtract, or perform further algebraic operations on fractions.

在 A-Level 爱德思考试中，最终答案通常要求写成最简精确形式。有理化后的分母被视为标准形式，并且更便于比较分数、加减分数或进行后续的代数运算。

For example, it is easier to see that 1/√2 = √2/2 is approximately 0.707 than to work with the reciprocal form directly. It also allows exact expressions to be combined cleanly.

例如，理解 1/√2 = √2/2 约为 0.707 比直接处理倒数形式更容易。它还能让精确表达式更清晰地合并。

3. Rationalising a Simple Surd Denominator | 单一根式分母的有理化

When the denominator is a single surd such as √a, multiply both the numerator and denominator by √a. This uses the fact that √a × √a = a, so the denominator becomes rational.

当分母是单个根式如 √a 时，将分子和分母同时乘以 √a。利用 √a × √a = a，分母就变为有理数。

1/√a = (1 × √a)/(√a × √a) = √a/a

For example, 1/√2 = √2/2 and 3/√5 = 3√5/5.

例如，1/√2 = √2/2，3/√5 = 3√5/5。

If the denominator is a multiple of a surd, such as 2√3, still multiply by √3. Thus 5/(2√3) = 5√3/(2 × 3) = 5√3/6.

如果分母是根式的倍数，例如 2√3，仍然乘以 √3。因此 5/(2√3) = 5√3/(2 × 3) = 5√3/6。

4. The Key Identity: Difference of Two Squares | 关键恒等式：平方差

For binomial denominators, the most important identity is the difference of two squares: (a + b)(a − b) = a² − b². When applied to surds, this removes the square-root terms from the product.

对于二项式分母，最重要的恒等式是平方差公式：(a + b)(a − b) = a² − b²。将其应用于根式时，可以消去乘积中的平方根项。

(√a + √b)(√a − √b) = (√a)² − (√b)² = a − b

More generally, for a denominator of the form p + q√r, its conjugate is p − q√r, and the product is p² − q²r because (q√r)² = q²r.

更一般地，对于形如 p + q√r 的分母，其共轭式为 p − q√r，它们的乘积为 p² − q²r，因为 (q√r)² = q²r。

5. Rationalising Binomial Denominators | 二项式分母的有理化

A binomial denominator is one with two terms, such as a + b√c or a − b√c. To rationalise it, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a + b√c is a − b√c, and the conjugate of a − b√c is a + b√c.

二项式分母是含有两项的分母，例如 a + b√c 或 a − b√c。要对其有理化，需将分子和分母同时乘以分母的共轭式。a + b√c 的共轭式是 a − b√c，a − b√c 的共轭式是 a + b√c。

1/(a + b√c) = (a − b√c)/(a² − b²c)

The denominator becomes a² − b²c, which is rational. Remember that the numerator must be multiplied by the same conjugate to keep the fraction equivalent.

分母变为 a² − b²c，这是有理数。请记住，分子必须乘以相同的共轭式，以保持分数的值不变。

6. Worked Example: Denominator a + √b | 例题：分母为 a + √b

Rationalise 1/(2 + √3). The conjugate of 2 + √3 is 2 − √3, so multiply the numerator and denominator by 2 − √3.

有理化 1/(2 + √3)。2 + √3 的共轭式是 2 − √3，因此将分子和分母同时乘以 2 − √3。

1/(2 + √3) = (2 − √3)/((2)² − (√3)²) = (2 − √3)/(4 − 3) = 2 − √3

Another example is 7/(3 + √5). Multiply by 3 − √5: 7(3 − √5)/(9 − 5) = (21 − 7√5)/4.

另一个例子是 7/(3 + √5)。乘以 3 − √5：7(3 − √5)/(9 − 5) = (21 − 7√5)/4。

7. Worked Example: Denominator a − √b and Sign Change | 例题：分母为 a − √b 及符号变化

When the denominator is a − √b, use a + √b as the conjugate. For example, rationalise 5/(√7 − 2). The conjugate is √7 + 2.

当分母为 a − √b 时，使用 a + √b 作为共轭式。例如，有理化 5/(√7 − 2)，共轭式为 √7 + 2。

5/(√7 − 2) = 5(√7 + 2)/((√7)² − 2²) = (5√7 + 10)/(7 − 4) = (5√7 + 10)/3

For an expression such as (√2 + 1)/(√2 − 1), multiply by √2 + 1. The numerator becomes (√2 + 1)² = 2 + 2√2 + 1 = 3 + 2√2, and the denominator becomes 2 − 1 = 1, so the result is 3 + 2√2.

对于诸如 (√2 + 1)/(√2 − 1) 的表达式，乘以 √2 + 1。分子变为 (√2 + 1)² = 2 + 2√2 + 1 = 3 + 2√2，分母变为 2 − 1 = 1，因此结果为 3 + 2√2。

8. Rationalising More Complex Expressions | 更复杂表达式的有理化

When the denominator contains two different surds, the same conjugate method applies. For example, rationalise 2/(√3 + √2). The conjugate is √3 − √2.

当分母含有两个不同的根式时，同样使用共轭式方法。例如，有理化 2/(√3 + √2)，共轭式为 √3 − √2。

2/(√3 + √2) = 2(√3 − √2)/((√3)² − (√2)²) = 2(√3 − √2)/(3 − 2) = 2√3 − 2√2

You can also rationalise fractions where both numerator and denominator contain surds. For example, (1 + √2)/(2 + √2) is rationalised by multiplying by 2 − √2. The numerator expands to 2 − √2 + 2√2 − 2 = √2, and the denominator becomes 4 − 2 = 2, giving √2/2.

你也可以对分子和分母都含有根式的分数进行有理化。例如，(1 + √2)/(2 + √2) 的有理化方法是乘以 2 − √2。分子展开为 2 − √2 + 2√2 − 2 = √2，分母变为 4 − 2 = 2，结果为 √2/2。

9. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One common mistake is multiplying only the denominator by the conjugate and forgetting to multiply the numerator. This changes the value of the fraction. Always multiply the numerator and denominator by the same expression.

一个常见错误是只将分母乘以共轭式，却忘记乘以分子。这会改变分数的值。务必让分子和分母同时乘以相同的表达式。

Another mistake is selecting the wrong conjugate. If the denominator is a + √b, multiplying by a + √b does not rationalise the denominator, because (a + √b)² = a² + 2a√b + b still contains a surd term.

另一个错误是选错共轭式。如果分母是 a + √b，乘以 a + √b 并不能使分母有理化，因为 (a + √b)² = a² + 2a√b + b 仍然含有根式项。

Students also sometimes square terms incorrectly. Remember that (p + q√r)(p − q√r) = p² − q²r, not p² − qr. Also, (√a + √b)² = a + 2√ab + b, not a + b.

学生有时还会错误地平方各项。请记住 (p + q√r)(p − q√r) = p² − q²r，而不是 p² − qr。此外，(√a + √b)² = a + 2√ab + b，而不是 a + b。

10. Exam-Style Practice and Answers | 真题风格练习与答案

Use the table below to practise rationalising denominators. Cover the answers first, then check your working.

请使用下表中的练习来训练分母有理化。先遮住答案，再核对自己的解题过程。

Question	Answer
Rationalise 4/√7	4√7/7
Rationalise 5/(2√3)	5√3/6
Rationalise 3/(2 − √5)	−6 − 3√5
Simplify (√3 + 1)/(√3 − 1)	2 + √3
Rationalise √6/(√3 + √2)	3√2 − 2√3

In each case, the working should show multiplication by the appropriate conjugate, correct expansion of the denominator using the difference of two squares, and simplification of the final surd expression.

每一题都应展示乘以恰当的共轭式、利用平方差公式正确展开分母，以及化简最终的根式表达式。

11. Summary and Key Points | 总结与要点

The key points for rationalising denominators are:

有理化分母的要点如下：

For a single surd √a, multiply by √a/√a.
对于单个根式 √a，乘以 √a/√a。
For a binomial p + q√r, multiply by its conjugate p − q√r.
对于二项式 p + q√r，乘以它的共轭式 p − q√r。
Use the difference of two squares: (p + q√r)(p − q√r) = p² − q²r.
运用平方差公式：(p + q√r)(p − q√r) = p² − q²r。
Always multiply both numerator and denominator, and simplify the final answer fully.
务必同时乘以分子和分母，并完整化简最终答案。

Practising these steps regularly will make rationalising denominators quick and reliable in Edexcel A-Level Mathematics exams.

定期练习这些步骤，将使分母有理化在爱德思 A-Level 数学考试中变得快速且可靠。

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Rationalising Denominators | 有理化分母