📚 Some Algebraic Rules | 一些代数法则
Algebraic rules form the backbone of mathematical manipulation in the IB Diploma Programme. From simplifying expressions to solving equations, a firm grasp of these foundational principles allows you to work efficiently and accurately. This article revisits key algebraical rules, presenting each concept in a structured manner with paired English and Chinese explanations to support bilingual learners.
代数法则是 IB 文凭课程中数学运算的基石。从化简表达式到解方程，牢固掌握这些基本原理能让你高效且准确地解题。本文以结构化的方式回顾关键的代数法则，并配以中英双语解释，为双语学习者提供支持。
1. Exponent Multiplication and Division | 指数的乘除法则
⁻ⁿ, provided a ≠ 0. These rules are used constantly to simplify monomials and polynomials.
同底数幂相乘时，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。相除时指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ（其中 a ≠ 0）。这些法则经常用于化简单项式与多项式。
2. Power of a Power and Power of a Product | 幂的乘方与积的乘方
factor: (ab)ⁿ = aⁿ bⁿ. These rules help break down complex expressions involving brackets.
幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐⁿ。积的乘方，将指数分配给每个因式：(ab)ⁿ = aⁿ bⁿ。这些法则有助于分解含有括号的复杂表达式。
3. Zero and Negative Exponents | 零指数与负指数
Any non‑ ⁿ. Understanding this allows you to rewrite expressions without negative indices, a common requirement in IB exam simplification tasks.
任何非零底数的零次幂等于 1：a⁰ = 1（a ≠ 0）。负指数表示倒数：a⁻ⁿ = 1 / aⁿ。掌握这一点可以让你重写不含负指数的表达式，这是 IB 考试化简题中的常见要求。
4. Fractional Exponents and Radicals | 分数指数与根式
An exponent of 1/n corresponds to the n‑th root: a1/n = ⁿ√a. More generally, am/n = (ⁿ√a)ᵐ = ⁿ√aᵐ. This connection between powers and roots is essential when working with surds or solving exponential equations.
指数 1/n 对应 n 次方根：a1/n = ⁿ√a。更一般地，am/n = (ⁿ√a)ᵐ = ⁿ√aᵐ。在处理根式或解指数方程时，这种幂与根的联系至关重要。
5. Logarithm Rules | 对数法则
exponential function. Three key rules are: logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x – logₐ y, and logₐ(xⁿ) = n logₐ x. The change‑of‑base formula logₐ b = log_c b / log_c a is also widely used.
对数是指数函数的逆运算。三条关键法则是：logₐ(xy) = logₐ x + logₐ y，logₐ(x/y) = logₐ x – logₐ y，以及 logₐ(xⁿ) = n logₐ x。换底公式 logₐ b = log_c b / log_c a 也被广泛使用。
6. Expanding Brackets (Distributive Law) | 去括号（分配律）
)(c + d) = ac + ad + bc + bd. Special products like (a + b)² = a² + 2ab + b² and (a + b)(a – b) = a² – b² appear frequently and should be recognised instantly.
单项式乘括号：a(b + c) = ab + ac。对于两个二项式，使用 FOIL 方法：(a + b)(c + d) = ac + ad + bc + bd。特殊乘积如 (a + b)² = a² + 2ab + b² 和 (a + b)(a – b) = a² – b² 经常出现，需能快速识别。
7. Factorisation Techniques | 因式分解技巧
he greatest common factor (GCF), grouping terms, and recognising difference of squares: a² – b² = (a + b)(a – inding numbers p and q that sum to the coefficient of x and multiply to the constant term.
将分配律反向运用即可分解因式。常用方法包括提取最大公因式、分组分解，以及识别平方差公式：a² – b² = (a + b)(a – b)。二次三项式通常可通过寻找两个数 p 和 q（它们的和为 x 的系数、积为常数项）分解为 (x + p)(x + q)。
8. Simplifying Algebraic Fractions | 化简代数分式
Algebraic fractions follow the same rules as numerical fractions. To add or subtract, find a common denominator. To multiply, multiply numerators and denominators separately. To divide, multiply by the reciprocal. Always factor numerators and denominators first, then cancel common factors, noting any restrictions (values that make the denominator zero).
代数分式的运算规则与数字分式相同。加减时先通分；相乘时分子乘分子、分母乘分母；相除时乘以倒数。务必先对分子分母进行因式分解，再约去公因式，并注明限制条件（使分母为零的值）。
9. Solving Linear Equations: Transposition | 解线性方程：移项
ble, you may add, subtract, multiply, or divide both sides of the equation by the same non‑zero quantity. This is the principle of balancing. For example, ax + b = c leads to x = (c – b)/a. Always check your solution by substituting it back into the original equation.
为了解出变量，可在方程两边同时加上、减去、乘以或除以同一个非零量，这就是方程平衡原理。例如 ax + b = c 可化为 x = (c – b)/a。最后务必把解代入原方程进行检验。
10. Properties of Inequalities | 不等式的性质
Inequalities behave like equations except when multiplying or dividing by a negative number, which reverses the inequality sign. For instance, –2x > 6 becomes x < –3. Compound inequalities can be split into two separate statements, and solutions are often represented on a number line or using interval notation.
不等式的性质与等式类似，但当两边同乘或同除以一个负数时，不等号方向要改变。例如 –2x > 6 化为 x < –3。复合不等式可以拆分成两个单独的不等式，解集常用数轴或区间符号表示。
11. Absolute Value Equations and Inequalities | 绝对值方程与不等式
The absolute value |x| is defined as the distance from zero on the number line. |x| = a (a > 0) gives x = a or x = –a. For inequalities, |x| < a means –a < x < a, while |x| > a means x < – or x > a. These geometric interpretations help solve more complex absolute value problems.
绝对值 |x| 在数轴上表示到零点的距离。|x| = a（a > 0）的解为 x = a 或 x = –a。对于不等式，|x| < a 表示 –a < x < a，而 |x| > a 表示 x < –a 或 x > a。这种几何解释有助于解决更复杂的绝对值问题。
12. Rearranging Formulae and Substitution | 公式变形与代入法
Rearranging a formula involves treating all variables except the subject as constants and using inverse operations to isolate the desired variable. Substitution means replacing a variable with a given expression, which often simplifies an equation or leads to a factorable form. Both skills are central to IB problem‑solving, especially in functions and equations.
公式变形是指将除目标变量外的所有变量视为常数，运用逆运算把目标变量解出来。代入法则是用给定的表达式替换变量，通常可以简化方程或化为可分解的形式。这两种技能在 IB 解题中十分关键，尤其是在函数与方程专题中。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图