📚 Year 8 Cambridge Advanced Mathematics: Quick Reference Formula & Theorem Handbook | 剑桥八年级进阶数学：公式定理速查手册
Welcome to your Year 8 Cambridge Advanced Mathematics quick reference handbook. This guide compiles all the essential formulas, rules, and theorems you need to master the Stage 8 curriculum. Use it as a handy companion for homework, revision, and exam preparation, ensuring you have every key concept at your fingertips.
欢迎使用剑桥八年级进阶数学速查手册。本指南汇总了掌握八年级课程所需的所有基本公式、法则和定理，方便你在完成作业、复习和备考时快速查阅，随时掌握每一个核心概念。
1. Index Laws | 指数定律
Product rule: When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.
乘积法则：同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。
Quotient rule: : When dividing powers with the same base, subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (where a ≠ 0).
商法则：同底数幂相除，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ（a ≠ 0）。
Power of a power: When raising a power to another exponent, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ.
幂的乘方：幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐⁿ。
Zero exponent: Any non-zero number raised to the power 0 equals 1: a⁰ = 1 (a ≠ 0).
零指数：任何非零数的零次幂都等于1：a⁰ = 1（a ≠ 0）。
Negative exponent: A negative exponent indicates the reciprocal: a⁻ⁿ = 1 / aⁿ.
负指数：负指数表示该数的倒数：a⁻ⁿ = 1 / aⁿ。
2. Expanding Brackets and Factorising | 展开与因式分解
Single bracket expansion: Multiply the term outside by each term inside: a(b + c) = ab + ac.
单项式乘括号：用括号外的项乘以括号内的每一项：a(b + c) = ab + ac。
Double bracket expansion: Use the FOIL method to expand (a + b)(c + d): multiply First, Outer, Inner, Last, then sum: ac + ad + bc + bd.
两括号相乘：使用首外内尾法展开 (a + b)(c + d)：分别乘首项、外项、内项、尾项，再相加：ac + ad + bc + bd。
Factorising: side the bracket. For example, 2x + 6 = 2(x + 3). Always expand back to check your answer.
因式分解：找出各项的最大公因式（HCF），将其放在括号外。例如，2x + 6 = 2(x + 3)。分解后一定要乘开验证。
3. Linear Equations and Inequalities | 线性方程与不等式
Solving linear equations: : Isolate the variable by performing inverse operations on both sides. For example, 2x + 5 = 13 → subtract 5, then divide by 2 → x = 4.
解线性方程：通过等号两边同时进行逆运算来分离变量。例如：2x + 5 = 13 → 两边减5，再除以2 → x = 4。
Solving inequalities: ultiplying or dividing by a negative number. For example, −3x < 9 → divide by −3 → x > −3.
解不等式：方法与解方程类似，但要牢记：当同乘以或除以一个负数时，不等号方向必须改变。例如，−3x < 9 → 除以−3 → x > −3。
Representing on a number line: Use an open circle for < or >, and a filled circle for ≤ or ≥.
在数轴上表示：用于 < 或 > 的不等式时用空心圆，用于 ≤ 或 ≥ 时用实心圆。
4. Sequences: The nth Term | 序列：第 n 项
Arithmetic sequence: onsecutive terms. The nth term formula is: T(n) = a + (n − 1)d, where a is the first term.
等差数列：相邻两项的差为常数 d 的数列。第 n 项公式为：T(n) = a + (n − 1)d，其中 a 为首项。
Simplified form: The formula can also be written as T(n) = dn + c, where c = a − d.
简化形式：公式也可写作 T(n) = dn + c，其中 c = a − d。
Example: For the sequence 4, 9, 14, 19, ..., d = 5 and a = 4, so T(n) = 4 + (n − 1) × 5 = 5n − 1.
示例：对于数列 4, 9, 14, 19, ...，公差 d = 5，首项 a = 4，因此 T(n) = 4 + (n − 1) × 5 = 5n − 1。
Finding terms: Substitute n = 1, 2, 3, ... into the nth term to generate the sequence.
求项：将 n = 1, 2, 3, ... 代入第 n 项公式即可生成数列。
5. Straight Line Graphs | 直线图像
Equation of a straight line: ent (slope) and c represents the y-intercept (the point where the line crosses the y-axis).
直线方程：y = mx + c，其中 m 代表斜率，c 代表 y 轴截距（直线与 y 轴的交点）。
Gradient formula: ise / run = (y₂ − y₁) / (x₂ − x₁), using any two points (x₁, y₁) and (x₂, y₂) on the line.
斜率公式：m = 纵差 / 横差 = (y₂ − y₁) / (x₂ − x₁)，可以使用直线上任意两点 (x₁, y₁) 和 (x₂, y₂) 计算。
Plotting a graph: Create a table of values for x and y, plot the coordinate pairs, and join them with a straight line.
绘制图像：先列出 x 和 y 的值表，描出坐标点，再用直线将它们连接起来。
Horizontal and vertical lines: y = k is a horizontal line; x = h is a vertical line.
水平线与竖直线：y = k 的图像是一条水平线；x = h 的图像是一条竖直线。
6. Angles and Parallel Lines | 角与平行线
Corresponding angles: When a transversal crosses two parallel lines, corresponding angles are equal. (Recognise the F-shape.)
同位角：当一条截线穿过两条平行线时，同位角相等。（观察 F 字形。）
Alternate interior angles: Alternate angles between the parallels are equal. (Recognise the Z-shape.)
内错角：平行线间的内错角相等。（观察 Z 字形。）
Co-interior angles: Interior angles on the same side of the transversal sum to 180°. (Recognise the C-shape.)
同旁内角：截线同侧的两个内角互补，和为180°。（观察 C 字形。）
Vertically opposite angles: When two lines intersect, opposite angles are equal.
对顶角：两直线相交时，对顶角相等。
Angles on a straight line: Sum to 180°. Angles around a point: Sum to 360°.
平角：和为180°。周角：绕一个点的所有角和为360°。
7. Polygons: Interior and Exterior Angles | 多边形：内角与外角
Sum of interior angles: For an n-sided polygon, the sum of interior angles = (n − 2) × 180°.
内角和：n 边形的内角和 = (n − 2) × 180°。
Sum of exterior angles: The sum of the exterior angles of any convex polygon (one exterior at each vertex) is always 360°.
外角和：任何凸多边形的外角和（每个顶点取一个外角）恒为360°。
Regular polygon interior angle: Each interior angle = (n − 2) × 180° ÷ n.
正多边形内角：每个内角 = (n − 2) × 180° ÷ n。
Regular polygon exterior angle: Each exterior angle = 360° ÷ n.
正多边形外角：每个外角 = 360° ÷ n。
8. Perimeter, Area and Volume Formulas | 周长、面积与体积公式
Perimeter and circumference: Rectangle: P = 2(l + w).
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