📚 Chapter Review 1: Algebraic Expressions | 第一章复习：代数表达式
Welcome to this Chapter Review 1 for Edexcel A-Level Pure Mathematics. This chapter focuses on algebraic expressions, index laws, surds and algebraic fractions. These skills underpin almost every later topic, so a secure grasp is essential for A/A* work.
欢迎阅读 Edexcel A-Level 纯数学第一章复习。本章重点为代数表达式、指数定律、根式与代数分式。这些技能是几乎所有后续主题的基础，扎实掌握对取得 A 或 A* 至关重要。
1. Core Index Laws | 核心指数定律
The three key index laws are: when multiplying like bases, add the exponents; when dividing, subtract the exponents; when raising a power to a power, multiply the exponents.
三条核心指数定律为：同底数幂相乘，指数相加；同底数幂相除，指数相减；幂的乘方，指数相乘。
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
You must also know that any non-zero base raised to the power 0 equals 1, so a⁰ = 1. Negative powers give reciprocals, and these two facts often appear together in simplification questions.
你还必须知道，任何非零底数的 0 次幂都等于 1，即 a⁰ = 1。负指数表示倒数，这两个事实经常同时出现在化简题中。
2. Expanding Brackets | 展开括号
Multiplying two binomials uses the distributive law: each term in the first bracket multiplies each term in the second. For example, (x + 3)(x − 2) gives x² − 2x + 3x − 6, which simplifies to x² + x − 6.
展开两个二项式使用分配律：第一个括号中的每一项分别乘以第二个括号中的每一项。例如，(x + 3)(x − 2) 得到 x² − 2x + 3x − 6，化简为 x² + x − 6。
Two special products are tested frequently. The perfect square expansion is (a+b)² = a² + 2ab + b². The difference of two squares is (a+b)(a−b) = a² − b². Recognising these patterns saves time and reduces sign errors.
两个特殊乘积经常被考查。完全平方展开式为 (a+b)² = a² + 2ab + b²。平方差公式为 (a+b)(a−b) = a² − b²。识别这些模式可以节省时间并减少符号错误。
When a bracket is multiplied by a negative term, every sign inside the bracket must change. For instance, −2(x − 5) = −2x + 10, not −2x − 10.
当括号乘以一个负数时，括号内的每一项符号都必须改变。例如，−2(x − 5) = −2x + 10，而不是 −2x − 10。
3. Factorising Linear and Quadratic Forms | 因式分解：一次与二次式
Always take out the highest common factor first. This is the fastest way to make a quadratic easier to factorise or to spot a difference of two squares. For example, 2x² + 6x = 2x(x + 3).
因式分解时首先提取最大公因式。这是让二次式更容易分解或发现平方差的最快方法。例如，2x² + 6x = 2x(x + 3)。
For quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b. Thus x² + 7x + 10 factorises as (x + 2)(x + 5) because 2 × 5 = 10 and 2 + 5 = 7.
对形如 x² + bx + c 的二次式，寻找两个数，它们的乘积为 c，和为 b。因此 x² + 7x + 10 因式分解为 (x + 2)(x + 5)，因为 2 × 5 = 10 且 2 + 5 = 7。
For quadratics with a leading coefficient not equal to 1, such as 2x² + 7x + 3, use a systematic method. Write it as 2x² + 6x + x + 3, then group: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
对于首项系数不为 1 的二次式，例如 2x² + 7x + 3，使用系统方法。将其写成 2x² + 6x + x + 3，然后分组：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。
4. Negative and Fractional Indices | 负指数与分数指数
A negative exponent indicates a reciprocal: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8. This rule works for algebraic terms too, so x⁻² = 1/x².
负指数表示倒数：a⁻ⁿ = 1/aⁿ。例如，2⁻³ = 1/2³ = 1/8。该规则也适用于代数项，因此 x⁻² = 1/x²。
A fractional exponent indicates a root: a¹⁄ⁿ = ⁿ√a. Therefore a¹⁄² = √a, and a¹⁄³ = ∛a. When the numerator is not 1, aᵐ⁄ⁿ = ⁿ√(aᵐ).
分数指数表示根式：a¹⁄ⁿ = ⁿ√a。因此 a¹⁄² = √a，a¹⁄³ = ∛a。当分子不为 1 时，aᵐ⁄ⁿ = ⁿ√(aᵐ)。
These rules can be combined in a single expression. For instance, 16⁻³⁄⁴ means 1 ÷ 16³⁄⁴. Since 16¹⁄⁴ = 2, we get 16³⁄⁴ = 2³ = 8, so 16⁻³⁄⁴ = 1/8.
这些规则可以组合在一个表达式中。例如，16⁻³⁄⁴ 表示 1 ÷ 16³⁄⁴。由于 16¹⁄⁴ = 2，得到 16³⁄⁴ = 2³ = 8，因此 16⁻³⁄⁴ = 1/8。
5. Surds and Rationalising Denominators | 根式与分母有理化
To simplify a surd, split the radicand into a square factor and a remaining factor. For example, √72 = √(36 × 2) = √36 × √2 = 6√2. The square factor must be the highest possible square number.
化简根式时，将被开方数拆为平方因子与剩余因子。例如，√72 = √(36 × 2) = √36 × √2 = 6√2。平方因子必须取尽可能大的平方数。
To rationalise a denominator like 1/√a, multiply numerator and denominator by √a. This gives √a/a. For a denominator such as 1/(2+√3), multiply by the conjugate 2−√3.
对形如 1/√a 的分母进行有理化，可将分子和分母同乘 √a。得到 √a/a。对于形如 1/(2+√3) 的分母，应乘以其共轭根式 2−√3。
For example, rationalising 1/(2+√3) gives (2−√3)/((2+√3)(2−√3)) = (2−√3)/(4−3) = 2−√3. The conjugate technique removes the square root from the denominator completely.
例如，将 1/(2+√3) 有理化得到 (2−√3)/((2+√3)(2−√3)) = (2−√3)/(4−3) = 2−√3。共轭技巧可以完全去除分母中的平方根。
6. Algebraic Fractions: Simplifying | 代数分式化简
Factorise the numerator and denominator first, then cancel common factors. Never cancel terms across addition or subtraction; only common factors can be removed.
化简代数分式时，先对分子和分母因式分解，再约去公因式。切勿在加减项之间约分；只有公因式才能约去。
For instance, (x² − 1)/(x + 1) = (x − 1)(x + 1)/(x + 1) = x − 1. This cancellation is valid only when x ≠ −1, because the original denominator cannot equal zero.
例如，(x² − 1)/(x + 1) = (x − 1)(x + 1)/(x + 1) = x − 1。这种约分仅在 x ≠ −1 时有效，因为原分母不能为零。
When adding or subtracting algebraic fractions, find a common denominator by factorising each denominator. For example, 1/(x−2) + 1/(x+2) = (x+2 + x−2)/((x−2)(x+2)) = 2x/(x²−4).
加减代数分式时，通过对每个分母因式分解来找到公分母。例如，1/(x−2) + 1/(x+2) = (x+2 + x−2)/((x−2)(x+2)) = 2x/(x²−4)。
7. Common Errors in Chapter 1 | 第一章常见错误
A very common error is writing (a+b)² = a² + b². This is wrong because the cross term 2ab is missing. The correct expansion is a² + 2ab + b², so (x+3)² equals x² + 6x + 9, not x² + 9.
一个非常常见的错误是写成 (a+b)² = a² + b²。这是错误的，因为漏掉了交叉项 2ab。正确的展开式是 a² + 2ab + b²，因此 (x+3)² 等于 x² + 6x + 9，而不是 x² + 9。
Another frequent mistake is misapplying index laws, such as writing aᵐ × aⁿ = aᵐⁿ instead of aᵐ⁺ⁿ. Similarly, aᵐ + aⁿ cannot be combined into a single power because the bases are terms, not factors.
另一个常见错误是误用指数定律，例如将 aᵐ × aⁿ 写成 aᵐⁿ，而不是 aᵐ⁺ⁿ。同样，aᵐ + aⁿ 不能合并为单一幂，因为它们是项而非因式。
When rationalising denominators, some students only multiply part of the denominator by the conjugate. The same factor must multiply both numerator and denominator so that the fraction’s value does not change.
在分母有理化时，一些学生只将分母的一部分乘以共轭根式。必须将分子和分母同时乘以相同的因子，这样分数的值才不会改变。
8. Worked Exam-Style Example | 考试风格例题详解
Question: Simplify (2x²y⁻³)³ × (x⁻¹y)², giving your answer with positive indices.
题目：化简 (2x²y⁻³)³ × (x⁻¹y)²
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