Exercise 1A: Laws of Indices and Surds | 练习 1A：指数律与根式

📚 Exercise 1A: Laws of Indices and Surds | 练习 1A：指数律与根式

Exercise 1A in the AQA A-Level Mathematics course focuses on the laws of indices and the manipulation of surds. This exercise builds algebraic fluency that will be used later in differentiation, integration, sequences and exponentials. You should aim to write answers as single powers or in simplest surd form.

AQA A-Level 数学课程中的练习 1A 重点考查指数律和根式的运算。该练习培养代数熟练度，后续学习微分、积分、数列和指数函数时都会用到。目标是把答案写成单一幂或最简根式。

1. What Exercise 1A Tests | 练习 1A 考查什么

The questions often look short, but they test whether you can choose the correct rule under time pressure. Always check whether bases are the same before combining powers.

题目通常看起来很短，但考查的是你在时间压力下能否选择正确法则。合并幂之前一定要先检查底数是否相同。

egative powers like x⁻², converting between fractional powers and roots, and rationalising denominators such as 1/√3.

典型技能包括化简 3x² × 5x⁴ 之类的表达式、使用 x⁻² 这样的负指数、在分数指数和根式之间转换，以及对 1/√3 这样的分母进行有理化。

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2. The Core Index Laws | 核心指数律

For a non-zero base a and positive integers m and n, the core laws are listed below. These laws only apply when the bases are the same for multiplication and division.

对于非零底数 a 和正整数 m、n，核心法则如下。乘法和除法中的这些法则仅在底数相同时适用。

Law	Symbol form	Meaning
Multiplying same base	aᵐ × aⁿ = aᵐ⁺ⁿ	Add the exponents
Dividing same base	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	Subtract the exponents
Power of a power	(aᵐ)ⁿ = aᵐⁿ	Multiply the exponents
Power of a product	(ab)ⁿ = aⁿbⁿ	Apply the power to each factor
Power of a quotient	(a/b)ⁿ = aⁿ/bⁿ	Apply the power to numerator and denominator

The first two laws explain why powers add when multiplying and subtract when dividing. The third law shows that a power raised to another power multiplies the exponents.

前两条法则解释了为什么相乘时指数相加、相除时指数相减。第三条法则表明幂的幂要把指数相乘。

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3. Zero and Negative Powers | 零指数与负指数

Zero exponent: a⁰ = 1 for any a ≠ 0. This comes from dividing aᵐ by aᵐ: aᵐ⁻ᵐ = a⁰ = 1.

零指数：对任意 a ≠ 0，a⁰ = 1。这可以通过 aᵐ ÷ aᵐ = aᵐ⁻ᵐ = a⁰ = 1 来理解。

= 1/aⁿ. A negative power means the reciprocal of the positive power, so 2⁻³ = 1/2³ = 1/8.

负指数：a⁻ⁿ = 1/aⁿ。负指数表示正指数的倒数，因此 2⁻³ = 1/2³ = 1/8。

You can also write 1/x⁵ as x⁻⁵ and change the sign of the exponent when moving a factor between the numerator and denominator.

你也可以把 1/x⁵ 写成 x⁻⁵，当因子在分子和分母之间移动时改变指数的符号。

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4. Fractional Powers and Roots | 分数指数与根式

Fractional powers connect indices to roots. The denominator of the fraction is the root, and the numerator is the power: a^1/2 = √a, a^1/3 = ∛a, and a^m/n = ⁿ√aᵐ = (ⁿ√a)ᵐ.

分数指数把幂与根式联系起来。分数的分母是根指数，分子是幂指数：a^1/2 = √a，a^1/3 = ∛a，而 a^m/n = ⁿ√aᵐ = (ⁿ√a)ᵐ。

Example: 8^2/3 = (∛8)² = 2² = 4. This is much easier than trying to square 8 first.

例如：8^2/3 = (∛8)² = 2² = 4。这比先算 8² 再开立方要简单得多。

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5. Simplifying Index Expressions | 化简指数表达式

+ (-1) = 2, giving x². The y-powers add: 2 + 4 = 6, giving y⁶. The final answer is 12x²y⁶.

要化简 6x³y² × 2x⁻¹y⁴，应分别处理系数、x 的幂和 y 的幂。系数部分是 6 × 2 = 12。x 的指数相加：3 + (-1) = 2，得到 x²。y 的指数相加：2 + 4 = 6，得到 y⁶。最终答案是 12x²y⁶。

roduct to a power, apply the power to each factor. For example, (2a³)² = 2² × (a³)² = 4a⁶.

当把乘积整体乘方时，要把幂分别作用到每个因式。例如 (2a³)² = 2² × (a³)² = 4a⁶。

For division, subtract the powers of like bases: (10a⁵b²) ÷ (2a²b) = 5a³b.

对于除法，同底数幂的指数相减：(10a⁵b²) ÷ (2a²b) = 5a³b。

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6. Surds in AQA Exercise 1A | AQA 练习 1A 中的根式

A surd is an irrational root, such as √2 or ∛5. In AQA Exercise 1A, you are usually expected to simplify surds by removing square factors from under the root.

根式是无理数根，如 √2 或 ∛5。在 AQA 练习 1A 中，通常要求通过去掉根号下的平方因子来化简根式。

Surds are exact values. Writing √8 as 2√2 keeps the number exact, whereas a decimal like 2.828... is only an approximation.

根式是精确值。把 √8 写成 2√2 保持精确，而 2.828... 这样的小数只是近似值。

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7. Simplifying Surds | 化简根式

To simplify √50, look for the largest square factor of 50. Since 50 = 25 × 2, we get √50 = √25 × 2 = √25 × √2 = 5√2.

要化简 √50，找出 50 的最大平方因子。因为 50 = 25 × 2，所以 √50 = √25 × 2 = √25 × √2 = 5√2。

Similarly, √72 = √36 × 2 = 6√2. The key rule is √ab = √a × √b when a and b are non-negative.

类似地，√72 = √36 × 2 = 6√2。关键法则是当 a、b 均为非负数时，√ab = √a × √b。

You can add like surds. For example, 3√2 + 5√2 = 8√2, but √3 + √5 cannot be combined because the radicands differ.

同类根式可以相加。例如 3√2 + 5√2 = 8√2，但 √3 + √5 不能合并，因为被开方数不同。

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8. Rationalising Denominators | 有理化分母

Rationalising the denominator means rewriting a fraction so the denominator is a rational number. For 1/√2, multiply the numerator and denominator by √2: 1/√2 = √2/2.

有理化分母就是把分数改写成使分母为有理数。对于 1/√2，将分子和分母同时乘以 √2：1/√2 = √2/2。

For a denominator like 3 + √2, multiply by its conjugate 3 − √2. For example, 5/(3 + √2) × (3 − √2)/(3 − √2) = 5(3 − √2)/(9 − 2) = (15 − 5√2)/7.

对于形如 3 + √2 的分母，乘以它的共轭 3 − √2。例如，5/(3 + √2) × (3 − √2)/(3 − √2) = 5(3 − √2)/(9 − 2) = (15 − 5√2)/7。

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9. Common Errors to Avoid | 常见错误与避免方法

ltiplying, not when adding. The expression x² + x³ cannot be simplified to a single power.

一个常见

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