📚 Surds: Simplification and Operations | 根式（无理数）的化简与运算
Surds are irrational numbers expressed in the form of square roots, cube roots, or other roots that cannot be simplified to a rational number. They appear frequently in A-Level mathematics, and mastering their simplification and operations is essential for solving equations, inequalities, and calculus problems.
根式是以平方根、立方根或其他根号形式表示的无理数，它们无法被化简为有理数。在A-Level数学中，根式频繁出现，掌握它们的化简与运算技巧对于解方程、不等式和微积分问题至关重要。
1. Basic Properties of Surds | 根式的基本性质
Before diving into operations, we must understand the fundamental properties of surds. For positive real numbers a and b, the following rules always hold: √a × b = √a × √b, and √a ÷ b = √a ÷ √b. These two properties form the backbone of all surd manipulations.
在深入运算之前，我们必须理解根式的基本性质。对于正实数 a 和 b，以下规则始终成立：√a × b = √a × √b，以及 √a ÷ b = √a ÷ √b。这两个性质构成了所有根式变形的基础。
Additionally, we must be careful with addition and subtraction: √a + √b does NOT equal √a + b. Similarly, √a − √b does NOT equal √a − b. Surds can only be combined when they have the same radicand (the number under the root sign), just like combining like terms in algebra.
此外，我们必须注意加减法：√a + √b 不等于 √a + b，√a − √b 也不等于 √a − b。只有被开方数（根号下的数）相同的根式才能合并，就像代数中合并同类项一样。
| Property | 性质 | Formula | 公式 |
| Multiplication | 乘法 | √a × √b = √ab |
| Division | 除法 | √a ÷ √b = √a/b |
| Square of a Surd | 根式的平方 | (√a)² = a |
| Perfect Square | 完全平方 | √a² = a (when a ≥ 0) |
2. Simplifying Surds into Simplest Form | 将根式化为最简形式
A surd is said to be in its simplest form when the number under the root sign has no perfect square factors (other than 1). To achieve this, we factor the radicand into a product of a perfect square and another factor, then apply √a × b = √a × √b.
当一个根号下的数字没有除1以外的完全平方因子时，该根式被称为最简形式。要实现这一点，我们将被开方数分解为一个完全平方数与另一个因数的乘积，然后应用 √a × b = √a × √b。
For example, simplify √72. Since 72 = 36 × 2, and 36 is a perfect square, we write √72 = √36 × 2 = √36 × √2 = 6√2. It is important to identify the largest perfect square factor to achieve the simplest form directly.
例如，化简 √72。由于 72 = 36 × 2，而 36 是完全平方数，我们写成 √72 = √36 × 2 = √36 × √2 = 6√2。关键在于找出最大的完全平方因子，以直接得到最简形式。
Let us practise with more examples: √50 = √25 × 2 = 5√2; √108 = √36 × 3 = 6√3; √125 = √25 × 5 = 5√5. Notice that the coefficient (the number outside the root) and the radicand (the number inside the root) must both be recorded accurately.
我们练习更多例子：√50 = √25 × 2 = 5√2；√108 = √36 × 3 = 6√3；√125 = √25 × 5 = 5√5。注意系数（根号外的数字）和被开方数（根号内的数字）都必须准确记录。
√72 = 6√2, √50 = 5√2, √108 = 6√3
3. Adding and Subtracting Surds | 根式的加减法
Addition and subtraction of surds follow the same principle as collecting like terms. Only surds with the identical radical part can be added or subtracted. For instance, 3√5 + 2√5 = 5√5, just as 3x + 2x = 5x in algebra.
根式的加减法遵循与合并同类项相同的原则。只有根号部分完全相同的根式才能进行加减。例如，3√5 + 2√5 = 5√5，就像代数中 3x + 2x = 5x 一样。
When surds appear to be different, we should first simplify each surd to its simplest form before attempting to combine them. For example, to evaluate √50 + √18, we first simplify: √50 = 5√2 and √18 = 3√2. Then we can add: 5√2 + 3√2 = 8√2.
当根式看起来不同时，我们应该先将每个根式化简为最简形式，然后再尝试合并。例如，计算 √50 + √18，我们首先化简：√50 = 5√2，√18 = 3√2。然后相加：5√2 + 3√2 = 8√2。
Note that √2 + √3 cannot be simplified further because √2 and √3 are not like surds. Always check whether simplification is possible before deciding that two surds cannot be combined.
注意 √2 + √3 无法继续化简，因为 √2 和 √3 不是同类根式。在判断两个根式是否可以合并之前，务必先检查是否可以进行化简。
√50 + √18 = 5√2 + 3√2 = 8√2
4. Multiplying Surds | 根式的乘法
Multiplying surds is straightforward: we multiply the coefficients together and multiply the radicands together. The rule is (p√a) × (q√b) = pq√ab. This works regardless of whether the radicands are the same or different.
根式的乘法非常直接：系数与系数相乘，被开方数与被开方数相乘。规则为 (p√a) × (q√b) = pq√ab。无论被开方数是否相同，这一规则都适用。
For example, (3√2) × (4√3) = 12√6. When multiplying a surd by itself, we use (√a)² = a, so (√7)² = 7. Furthermore, (2√5)² = 4 × 5 = 20 because we square both the coefficient and the radical part.
例如，(3√2) × (4√3) = 12√6。当根式自乘时，我们使用 (√a)² = a，所以 (√7)² = 7。此外，(2√5)² = 4 × 5 = 20，因为我们需要同时平方系数和根号部分。
After multiplying, always check whether the resulting surd can be simplified. For instance, (2√3) × (√6) = 2√18 = 2 × 3√2 = 6√2. Simplification at the end ensures our final answer is in the standard form.
乘法完成后，务必检查所得根式是否可以化简。例如，(2√3) × (√6) = 2√18 = 2 × 3√2 = 6√2。最后进行化简以确保最终答案处于标准形式。
5. Dividing Surds and Rationalising the Denominator | 根式的除法与分母有理化
Division of surds follows the rule √a ÷ √b = √a/b. However, a crucial skill in surd manipulation is rationalising the denominator. This means converting a fraction with a surd in the denominator into an equivalent fraction with a rational denominator.
根式的除法遵循规则 √a ÷ √b = √a/b。然而，根式变形中一项关键技能是分母有理化。这意味着将分母含有根式的分数转换为分母为有理数的等价分数。
To rationalise a denominator of the form √b, we multiply both the numerator and the denominator by √b. For example, to simplify 5/√3, we multiply by √3/√3, giving us 5√3/3. This technique eliminates the surd from the denominator.
要将形如 √b 的分母有理化，我们将分子和分母同时乘以 √b。例如，化简 5/√3，我们乘以 √3/√3，得到 5√3/3。这一技巧消除了分母中的根式。
5/√3 = (5 × √3) / (√3 × √3) = 5√3/3
For denominators of the form a + √b or a − √b, we use the difference of squares formula: (a + √b)(a − √b = a² − b. We multiply both numerator and denominator by the conjugate of the denominator.
对于形如 a + √b 或 a − √b 的分母，我们利用平方差公式：(a + √b)(a − √b) = a² − b。我们将分子和分母同时乘以分母的共轭表达式。
6. Rationalising with Conjugates | 用共轭表达式有理化
When the denominator is a binomial involving surds, such as 3 + √2, we multiply by the conjugate, which is 3 − √2. The product of a binomial and its conjugate is always rational, eliminating the surd from the denominator.
当分母是含根式的二项式时，例如 3 + √2，我们乘以它的共轭式 3 − √2。二项式与其共轭式的乘积始终是有理数，从而消除了分母中的根式。
Let us work through an example: simplify 7/(3 + √2). Multiply the numerator and denominator by the conjugate 3 − √2. The denominator becomes (3 + √2)(3 − √2) = 9 − 2 = 7. Thus, the fraction simplifies to 7(3 − √2)/7 = 3 − √2.
让我们通过一个例子来理解：化简 7/(3 + √2)。分子和分母同时乘以共轭式 3 − √2。分母变为 (3 + √2)(3 − √2) = 9 − 2 = 7。因此，该分数化简为 7(3 − √2)/7 = 3 − √2。
7/(3 + √2) = 7(3 − √2)/(9 − 2) = 3 − √2
This technique is particularly powerful when simplifying algebraic fractions that contain surds in the denominator, and it is frequently tested in both pure mathematics and mechanics contexts.
这一技巧在处理分母含根式的代数分式时特别有效，在纯数学和应用力学中都被频繁考查。
7. Expanding Brackets with Surds | 含根式的括号展开
kets with surds follows the same distributive law as ordinary algebra. For example, √2(3 + √5) = 3√2 + √10. Each term inside the brackets must be multiplied by the surd outside.
含根式的括号展开遵循与普通代数相同的分配律。例如，√2(3 + √5) = 3√2 + √10。括号内的每一项都必须与括号外的根式相乘。
When expanding two binomials containing surds, we use the FOIL method (First, Outer, Inner, Last). For instance, (√2 + 1)(√2 − 3) = √2 × √2 − 3√2 + √2 − 3 = 2 − 2√2 − 3 = −1 − 2√2.
当展开两个含根式的二项式乘积时，我们使用FOIL方法（首项、外项、内项、末项）。例如，(√2 + 1)(√2 − 3) = √2 × √2 − 3√2 + √2 − 3 = 2 − 2√2 − 3 = −1 − 2√2。
Care must be taken with signs. When the two binomials are conjugates of each other, the cross terms cancel out. For example, (√5 + 2)(√5 5 − 4 = 1. This identity is the basis for rationalising denominators with binomial surds.
需要注意符号问题。当两个二项式互为共轭式时，交叉项相互抵消。例如，(√5 + 2)(√5 − 2) = 5 − 4 = 1。这个恒等式是二项式根式分母有理化的基础。
(√5 + 2)(√5 − 2) = (√5)² − 2² = 5 − 4 = 1
8. Surds in Quadratic Equations | 二次方程中的根式
olve surds. For example, solving x² − 4x + 1 = 0 using the quadratic formula gives x = 2 ± √3.
根式经常作为二次方程的解出现。当判别式为正数但不是完全平方数时，根将包含根式。例如，使用求根公式解 x² − 4x + 1 = 0 得到 x = 2 ± √3。
re form: x² − 6x + 7 = (x − 3)² − 2. From this form, we can see that the roots are x = 3 ± √2. This connection between completing the square and surds is fundamental in coordinate geometry.
通常，我们需要将二次表达式写成配方法的形式：x² − 6x + 7 = (x − 3)² − 2。从这个形式可以看出根为 x = 3 ± √2。配方法与根式之间的联系在解析几何中至关重要。
When solving equations with surds, we should always verify our answers by substituting back into the original equation, especially after squaring both sides, as this operation can introduce extraneous solutions.
在解含根式的方程时，我们应该始终将答案代回原方程验证，特别是在两边平方后，因为这一操作可能会引入多余的解。
9. Nested Surds and Special Identities | 嵌套根式与特殊恒等式
A nested surd is an expression such as √a + √b. These can sometimes be simplified using the identity √x + 2√y = √a + √b, where a + b = x and ab = y. For example, √5 + 2√6 = √3 + √2, since 3 + 2 = 5 and 3 × 2 = 6.
嵌套根式是形如 √a + √b 的表达式。有时可以利用恒等式 √x + 2√y = √a + √b 进行化简，其中 a + b = x，ab = y。例如，√5 + 2√6 = √3 + √2，因为 3 + 2 = 5 且 3 × 2 = 6。
To verify this, we can square √3 + √2: (√3 + √2)² = 3 + 2 + 2√6 = 5 + 2√6. Taking the square root of both sides confirms the identity. This technique is sometimes tested in olympiad-style questions within A-Level further mathematics.
为了验证这一点，我们对 √3 + √2 平方：(√3 + √2)² = 3 + 2 + 2√6 = 5 + 2√6。对方程两边取平方根即可确认该恒等式。这一技巧有时会出现在进阶数学中奥林匹克风格的问题里。
√5 + 2√6 = √3 + √2
In general, for √a − 2√b, we seek the pattern (√m − √n)² = a − 2√b requires systematic trial and inspection of factor pairs rather than memorising formulas.
对于一般的 √a − 2√b，我们寻找模式 (√m − √n)² = a − 2√b，其中 m + n = a，mn = b。这种方法需要系统地尝试和检查因数对，而不是死记公式。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
A common mistake is treating √a + b as √a + √b. This is completely incorrect. For example, √9 + 16 = √25 = 5, but √9 + √16 = 3 + 4 = 7. Always remember that the square root of a sum is not the sum of square roots.
一个常见错误是将 √a + b 视为 √a + √b。这是完全错误的。例如，√9 + 16 = √25 = 5，但 √9 + √16 = 3 + 4 = 7。始终记住：和的开方不等于开方后的和。
Another frequent error is failing to simplify surds completely before performing operations. In an exam, marks are often awarded for the final simplified answer. A correct but unsimplified answer like √50 might not receive full credit when 5√2 is expected.
另一个常见错误是运算前未能将根式完全化简。在考试中，分数常常取决于最终化简后的答案。当期望答案为 5√2 时，写 √50 尽管正确却可能无法获得全部分数。
When rationalising the denominator, remember to multiply both the numerator and the denominator by the same quantity. Multiplying only the denominator changes the value of the fraction. Always check that your simplification maintains equivalence.
分母有理化时，请记住分子和分母必须乘以相同的量。只乘分母会改变分数的值。始终检查你的化简是否保持了等价性。
Finally, practise recognising perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Quick recognition of these numbers dramatically speeds up surd simplification in timed conditions.
最后，练习识别完全平方数：4、9、16、25、36、49、64、81、100、121、144、169、196、225。在限时条件下快速识别这些数字能大幅加快根式化简的速度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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