Mixed Exercise 1 Algebraic Expressions | 综合练习一 代数表达式

📚 Mixed Exercise 1 Algebraic Expressions | 综合练习一 代数表达式

Mixed Exercise 1 in the Edexcel AS/A Level Pure Mathematics course is not just a random set of questions. It brings together the core algebra techniques from the opening chapter: simplifying powers, expanding brackets, factorising, and manipulating surds. Strong performance here shows you have the foundations needed for quadratics, graphs, binomial expansion, and calculus later in the course.

在 Edexcel AS/A Level 纯数学课程中，综合练习一并不是一组随意拼凑的题目。它整合了开篇章节中的核心代数技能：幂的化简、去括号展开、因式分解以及根式运算。如果你能熟练完成这些题目，说明你已经掌握了后续二次函数、图像、二项式展开和微积分等章节所需的基础。

1. What Mixed Exercise 1 Covers | 综合练习一涵盖内容

Mixed Exercise 1 typically appears at the end of Chapter 1 of the Pearson Edexcel Pure Mathematics Year 1/AS book. It tests four main skill areas: applying index laws, expanding products of brackets, factorising linear and quadratic expressions, and simplifying or rationalising surds.

综合练习一通常出现在 Pearson Edexcel 纯数学 Year 1/AS 教材第 1 章末尾。它主要考查四个技能领域：应用指数法则、展开括号乘积、分解一次和二次表达式，以及化简根式或将分母有理化。

The questions are often labelled P, E, or E/P to show whether they target pure problem-solving, exam-style work, or mixed exam and problem-solving. This structure helps you check both fluency and deeper understanding.

题目通常标有 P、E 或 E/P，表示其目标是纯问题解决、考试题型，或考试与问题解决混合。这种结构有助于你同时检查熟练度和更深层的理解。

Many students find Mixed Exercise 1 straightforward at first, but small slips with signs, indices, or surds can quickly lose marks. Treat it as a diagnostic: if you can finish it accurately without notes, you are ready to move on.

许多学生一开始会觉得综合练习一很简单，但符号、指数或根式上的一点小失误就可能导致丢分。把它当作一次诊断：如果你能在不翻笔记的情况下准确完成，就说明你可以继续推进。

2. Index Laws and Simplification | 指数法则与化简

The first half of Mixed Exercise 1 often asks you to simplify expressions such as 2x³ × 3x⁵ or (4a²)³. You need to know the three key index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ.

综合练习一前半部分经常要求你化简如 2x³ × 3x⁵ 或 (4a²)³ 这样的表达式。你需要掌握三条关键指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，以及 (aᵐ)ⁿ = aᵐⁿ。

aᵐ × aⁿ = aᵐ⁺ⁿ    aᵐ ÷ aⁿ = aᵐ⁻ⁿ    (aᵐ)ⁿ = aᵐⁿ

For coefficients, multiply or divide the numbers first, then apply the index laws to the letter parts. For example, 2x³ × 3x⁵ gives 2 × 3 = 6 as the coefficient and x³ × x⁵ = x⁸, so the result is 6x⁸.

对于系数，先乘或除以数字部分，然后再对字母部分应用指数法则。例如，2x³ × 3x⁵ 的系数为 2 × 3 = 6，字母部分为 x³ × x⁵ = x⁸，因此结果是 6x⁸。

You should also remember that any non-zero base raised to the power 0 is 1. So 5x⁰ = 5 × 1 = 5. This rule often appears in mixed questions to test whether you understand the difference between coefficients and powers.

你还要记住，任何非零底数的 0 次幂都等于 1。因此 5x⁰ = 5 × 1 = 5。这一法则经常出现在混合题中，用来考查你是否理解系数与幂的区别。