📚 Surds | 无理根式
In A-Level Mathematics, surds are irrational roots that cannot be simplified into a rational number. They often appear as square roots, cube roots, and higher roots of integers. Surds are crucial for exact calculations, geometry, and advanced algebra. Understanding how to manipulate surds—simplifying, adding, multiplying, dividing, and rationalising denominators—is essential for success in Edexcel examinations.
在 A-Level 数学中，无理根式是指无法化简为有理数的无理根，通常以整数的平方根、立方根等形式出现。它们在精确计算、几何和高等代数中至关重要。掌握无理根式的各种运算——化简、加减、乘除和分母有理化——是 Edexcel 考试取得高分的关键。
1. What Are Surds? | 什么是无理根式
on containing a root symbol (√) that cannot be simplified to remove the root. For example, √2 and ∛5 are surds because they are irrational numbers. In contrast, √4 is not a surd because it simplifies to 2, a rational number. Surds are written in their exact form to avoid rounding errors. In Edexcel exams, leaving answers in surd form is often required unless stated otherwise.
无理根式是含有根号（√）且无法化简去掉根号的表达式。例如 √2 和 ∛5 都是无理根式，因为它们是无理数。相反，√4 不是无理根式，因为它可以化简为有理数 2。无理根式以精确形式书写以避免舍入误差。在 Edexcel 考试中，除非另有说明，通常要求将答案保留为无理根式。
2. Simplifying Surds | 简化无理根式
To simplify a surd, look for the largest square factor under the root. For example, √50 = √25×2 = √25 × √2 = 5√2. Always ensure the surd is expressed in the form a√b where a is an integer and b is square-free (no square factors other than 1). Use the rule √ab = √a × √b for positive a and b.
简化无理根式时，寻找根号下最大的平方因子。例如，√50 = √25×2 = √25 × √2 = 5√2。始终确保无理根式表示为 a√b 的形式，其中 a 为整数，b 为无平方因子（除 1 外无完全平方数因子）。运用公式 √ab = √a × √b（a 和 b 为正数）。
√a × b = √a × √b
Common square factors to recognise: 4, 9, 16, 25, 36, 49, 64, 81, 100. Practice simplifying surds like √72 = 6√2, √48 = 4√3, and √18 = 3√2.
常见的平方因子：4, 9, 16, 25, 36, 49, 64, 81, 100。练习简化如 √72 = 6√2，√48 = 4√3，√18 = 3√2。
3. Adding and Subtracting Surds | 无理根式的加减
Surds can be added or subtracted only when the 'like' surd part is identical, similar to collecting like terms in algebra. For example, 3√2 + 5√2 = 8√2. However, √2 + √3 cannot be simplified further because the radicands are different. Always simplify each surd first to check for like terms: √8 + √2 = 2√2 + √2 = 3√2.
无理根式只能在其根号部分完全相同（同类项）时进行加减运算，类似于代数中的合并同类项。例如，3√2 + 5√2 = 8√2。但 √2 + √3 无法进一步化简，因为根号内数字不同。务必首先化简每一个无理根式以检查是否为同类项：√8 + √2 = 2√2 + √2 = 3√2。
a√c ± b√c = (a ± b)√c
Remember that a√c and b√d unless c = d. This rule is fundamental for simplifying expressions and solving equations.
请记住，a√c 和 b√d 只有在 c = d 时才是同类项。这是化简表达式和解方程的基础规则。
4. Multiplying Surds | 无理根式的乘法
Multiplication of surds uses the property √a × √b = √ab. Multiply the coefficients (numbers outside the root) together and the radicands (numbers inside) together. For example, 2√3 × 4√5 = (2×4) × √3×5 = 8√15. If the product results in a surd that can be simplified, always reduce it: √6 × √2 = √12 = 2√3.
无理根式的乘法运用性质 √a × √b = √ab。将系数（根号外的数字）相乘，根号内的数字相乘。例如，2√3 × 4√5 = (2×4) × √3×5 = 8√15。如果乘积结果可以化简，一定要简化：√6 × √2 = √12 = 2√3。
Also, note that (√a)² = a, which is extremely useful when squaring surds. For instance, (3√2)² = 3² × (√2)² = 9 × 2 = 18.
另请注意，(√a)² = a，这对平方无理根式非常有用。例如，(3√2)² = 3² × (√2)² = 9 × 2 = 18。
When multiplying brackets, expand as usual: (√a + √b)(√a – √b) = a – b, a common method for rationalising denominators.
括号相乘时，正常展开：(√a + √b)(√a – √b) = a – b，这是分母有理化的常用方法。
5. Dividing Surds and Rationalising Denominators | 无理根式的除法与分母有理化
Dividing surds follows √a / √b = √a/b. For example, √18 / √2 = √18/2 = √9 d, rationalise the denominator by multiplying numerator and denominator by that surd. For instance, 5 / √3 = (5√3) / (√3 × √3) = (5√3) / 3. This ensures no surd appears in the denominator, which is standard form.
无理根式的除法遵循 √a / √b = √a/b。例如，√18 / √2 = √18/2 = √9 = 3。如果分母是单个无理根式，通过将分子和分母同时乘以该根式来使分母有理化。例如，5 / √3 = (5√3) / (√3 × √3) = (5√3) / 3。这样确保分母中不出现根式，符合标准形式。
a / √b = (a√b) / b
Always simplify the fraction after rationalising, if possible. Remember that Edexcel mark schemes often require rationalised denominators.
有理化后，如果可能，请简化分数。请记住，Edexcel 评分标准通常要求分母有理化。
6. Expanding Brackets with Surds | 含无理根式的括号展开
Expanding expressions like (a + √b)(c + √d) requires applying the distributive law (FOIL method). For example, (2 + √3)(1 + √3) = 2×1 + 2×√3 + √3×1 + √3×√3 = 2 + 2√3 + √3 + 3 = 5 + 3√3. Collect like terms and simplify surds where possible.
像 (a + √b)(c + √d) 这样的表达式展开需要运用分配律（FOIL 法）。例如，(2 + √3)(1 + √3) = 2×1 + 2×√3 + √3×1 + √3×√3 = 2 + 2√3 + √3 + 3 = 5 + 3√3。合并同类项并尽可能简化无理根式。
Special products are important: (√a + √b)(√a – √b) = a – b, and (p + √q)² = p² + 2p√q + q. Recognising patterns speeds up simplification.
特殊乘积很重要：(√a + √b)(√a – √b) = a – b，以及 (p + √q)² = p² + 2p√q + q。识别这些模式可以加快化简速度。
In Edexcel problems, expanding and simplifying often leads to expressions in the form p + q√r, which is the required final form.
在 Edexcel 考题中，展开和化简通常会得到 p + q√r 形式的表达式，这是要求的最终形式。
7. Rationalising Complex Denominators (Conjugate) | 复杂分母的有理化（共轭）
When the denominator is a binomial containing a surd, e.g., a + √b or a – √b, rationalise by multiplying numerator and denominator by the conjugate. The conjugate of a + √b is a – √b, and vice versa. For example, to rationalise 1/(2 + √3), multiply by (2 – √3)/(2 – √3): numerator = 2 – √3; denominator = 4 – 3 = 1, giving 2 – √3.
当分母是含有无理根式的二项式，如 a + √b 或 a – √b 时，通过乘以共轭式来有理化。a + √b 的共轭是 a – √b，反之亦然。例如，要使 1/(2 + √3) 有理化，乘以 (2 – √3)/(2 – √3)：分子 = 2 – √3；分母 = 4 – 3 = 1，得到 2 – √3。
(a + √b)(a – √b) = a² – b
Similarly, for expressions like (√5 + √2)/(√5 – √2), multiply by (√5 + √2)/(√5 + √2) to get (5 + 2√10 + 2)/(5 – 2) = (7 + 2√10)/3. Practice leads to fluency.
类似地，对于像 (√5 + √2)/(√5 – √2) 的表达式，乘以 (√5 + √2)/(√5 + √2) 得到 (5 + 2√10 + 2)/(5 – 2) = (7 + 2√10)/3。多加练习便可熟练。
8. Solving Equations with Surds | 解含有无理根式的方程
Equations involving surds often require isolating the surd term and then squaring both sides. For example, √x + 3 = x – 1. Square both sides: x + 3 = (x – 1)² → x + 3 = x² – 2x + 1 → 0 = x² – 3x – , as squaring can introduce false roots. Always verify solutions in the original equation.
含有无理根式的方程通常需要先隔离根式项，然后两边平方。例如，√x + 3 = x – 1。两边平方：x + 3 = (x – 1)² → x + 3 = x² – 2x + 1 → 0 = x² – 3x – 2。求解二次方程，然后检查增根，因为平方可能会引入假根。始终将解代入原方程验证。
For equations with two surds, you may need to isolate one, square, simplify, isolate the remaining surd, and square again. For instance, √x + √x+3 = 3. Systematic steps are essential.
对于含有两个根式的方程，可能需要先隔离一个根式，平方，化简，再隔离剩余的根式，再次平方。例如，√x + √x+3 = 3。系统的解题步骤至关重要。
9. Surds and Exact Values | 无理根式与精确值
s provide exact values that cannot be expressed as finite decimals. For example, sin 60° = √3/2, the diagonal of a unit square is √2, and the golden ratio involves √5.
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