AQA GCSE Maths: Roots and Integer Indices Past Paper | 根与整数指数真题精练

AQA GCSE Mathematics — Roots and Integer Indices is a foundational topic that appears consistently across both Foundation and Higher tier papers. From simplifying expressions like 3³ × 3⁵ to working with fractional and negative indices, these skills are essential for algebra, surds, and beyond. This past paper (5 pages) from Physics & Maths Tutor gives you authentic exam-style practice with calculator and non-calculator questions.

AQA GCSE 数学 — 根与整数指数 是一个贯穿基础卷和高级卷的基础主题。从简化 3³ × 3⁵ 到处理分数指数和负指数，这些技能对代数、根式及更高层次的学习都至关重要。这份来自 Physics & Maths Tutor 的 5 页真题，为你提供真实的考试风格练习，涵盖计算器和非计算器题型。

Key Knowledge Points / 核心知识点

1. Index Laws — The Foundation / 指数法则 — 基础篇

The three fundamental rules: Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ (add the powers when the base is the same). Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract the powers). Power of a power: (aᵐ)ⁿ = aᵐⁿ (multiply the powers). These three rules underpin every indices question at GCSE.

三条基本法则：乘法：aᵐ × aⁿ = aᵐ⁺ⁿ（底数相同时指数相加）。除法：aᵐ ÷ aⁿ = aᵐ⁻ⁿ（指数相减）。幂的幂：(aᵐ)ⁿ = aᵐⁿ（指数相乘）。这三条法则支撑了 GCSE 考试中的每一道指数题。

2. Zero and Negative Indices / 零指数与负指数

Zero index: a⁰ = 1 for any non-zero a. Think of it as aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰ = 1. Negative index: a⁻ⁿ = 1/aⁿ. A negative index means “reciprocal” — it moves the term from numerator to denominator (or vice versa). Example: 2⁻³ = 1/2³ = 1/8.

零指数：对于任何非零 a，a⁰ = 1。可以理解为 aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰ = 1。负指数：a⁻ⁿ = 1/aⁿ。负指数意味着”取倒数” — 它将该项从分子移到分母（或反之）。例如：2⁻³ = 1/2³ = 1/8。

3. Fractional Indices = Roots / 分数指数 = 根式

Denominator = root: a^(1/n) = ⁿ√a (the nth root). Numerator = power: a^(m/n) = (ⁿ√a)ᵐ or ⁿ√(aᵐ) — both interpretations are correct. AQA frequently tests whether you can evaluate expressions like 8^(2/3): first cube root of 8 = 2, then square: 2² = 4. Or: 8² = 64, then cube root: ³√64 = 4. Either path works.

分母 = 根指数：a^(1/n) = ⁿ√a（n 次方根）。分子 = 幂次：a^(m/n) = (ⁿ√a)ᵐ 或 ⁿ√(aᵐ) — 两种理解都正确。AQA 经常考查你能否计算如 8^(2/3) 的表达式：先求 8 的立方根 = 2，再平方：2² = 4。或者：8² = 64，再开立方根：³√64 = 4。两种路径都对。

4. Simplifying Expressions with Indices / 化简含指数的表达式

When simplifying 5x³y² × 3x⁴y³: multiply the coefficients (5 × 3 = 15), then apply the multiplication law to each variable — x³ × x⁴ = x⁷, y² × y³ = y⁵. Final answer: 15x⁷y⁵. Always handle numbers and variables separately, then combine.

化简 5x³y² × 3x⁴y³ 时：先将系数相乘 (5 × 3 = 15)，再对每个变量应用乘法法则 — x³ × x⁴ = x⁷，y² × y³ = y⁵。最终答案：15x⁷y⁵。始终分别处理数字和变量，然后合并。

5. Calculator Skills & Estimation / 计算器技巧与估算

AQA often asks you to use a calculator for complex roots (e.g., ⁴√81 or ∛27) and then check with estimation. For √50: 7² = 49 and 8² = 64, so √50 ≈ 7.07 — your calculator answer should be close to this. Estimation catches keystroke errors and earns method marks even if the final answer is wrong.

AQA 经常要求用计算器计算复杂根式（如 ⁴√81 或 ∛27），然后 用估算验证。以 √50 为例：7² = 49，8² = 64，所以 √50 ≈ 7.07 — 你的计算器答案应接近这个值。估算可以发现按键错误，即使最终答案错了也能拿到方法分。

Study Tips / 学习建议

• Memorise the three index laws cold: Write them on a flashcard — aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. These three cover 80% of indices questions.
• Practice fractional-to-root conversion: Train yourself to see a^(1/2) as √a and a^(1/3) as ³√a instantly. Speed matters in the non-calculator paper.
• Work backwards to verify: After simplifying 6¹⁰ ÷ 6² = 6⁸, multiply back — does 6⁸ × 6² = 6¹⁰? Yes. This habit catches sign errors.
• Use past papers under timed conditions: This 5-page paper from PMT is ideal for a 15-minute focused practice session. Mark yourself and review every mistake.

• 牢记三条指数法则：写在闪卡上 — aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。这三条覆盖了 80% 的指数考题。
• 练习分数指数与根式的转换：训练自己瞬间将 a^(1/2) 看作 √a，将 a^(1/3) 看作 ³√a。在非计算器卷中速度至关重要。
• 反向验证：化简 6¹⁰ ÷ 6² = 6⁸ 后，乘回去 — 6⁸ × 6² = 6¹⁰？正确。这个习惯能捕捉符号错误。
• 计时做真题：这份来自 PMT 的 5 页真题非常适合 15 分钟限时训练。自己批改并复习每一个错误。

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