Negative Indices | 负指数

📚 Negative Indices | 负指数

Negative indices (also called negative exponents) are a way of expressing reciprocals in index notation. If you see a power with a negative sign, it tells you to take the reciprocal of the base and then raise it to the corresponding positive power. Mastering this concept is essential for simplifying algebraic expressions, evaluating numerical powers, and solving equations in IGCSE Edexcel Mathematics.

负指数（也称为负的幂）是一种用指数形式表示倒数的方式。当你看到一个带负号的幂时，这意味着你要先取底数的倒数，然后再将其进行相应正指数的乘方运算。掌握负指数对于化简代数表达式、计算数值幂以及求解方程在IGCSE Edexcel数学中至关重要。

1. What Are Negative Indices? | 什么是负指数？

A negative index does not mean the number is negative; it indicates the reciprocal of the base raised to a positive power. For example, 2⁻³ is equal to 1/(2³), not −8.

负指数并不意味着数字本身为负；它表示底数取倒数后再进行正指数运算。例如，2⁻³ 等于 1/(2³)，而不是 −8。

lows naturally from the laws of indices when we consider dividing powers of the same base.

一般而言，任何非零数 a 的 −n 次方都满足法则 a⁻ⁿ = 1 / aⁿ。这一规定是基于同底数幂的除法法则自然推导出来的。

2. The Fundamental Rule: a⁻ⁿ = 1 / aⁿ | 基本法则：a⁻ⁿ = 1 / aⁿ

exponent n, find the value of aⁿ, and then take its reciprocal. Conversely, 1 / a⁻ⁿ = aⁿ.

你需要记住的最重要公式是：当 a ≠ 0 时，a⁻ⁿ = 1 / aⁿ。这意味着你取正指数 n，算出 aⁿ 的值，然后取它的倒数。反之，1 / a⁻ⁿ = aⁿ。

Here are some simple numerical examples: 3⁻² = 1 / 3² = 1/9 10⁻¹ = 1/10 (−2)⁻³ = 1 / (−2)³ = 1 / (−8) = −1/8 gative after taking the reciprocal because the odd power of a negative number is negative.

以下是几个简单的数值示例： 3⁻² = 1 / 3² = 1/9 10⁻¹ = 1/10 (−2)⁻³ = 1 / (−2)³ = 1 / (−8) = −1/8 注意，如果底数是负数且负指数为奇数，取倒数后结果仍为负，因为负数的奇次幂为负。

3. Negative Indices with Fractions | 分数的负指数

lies. Essentially, the reciprocal of the fraction is taken, and then it is raised to the positive power n. This is a very common simplification skill in IGCSE exams.

当一个分数整体进行负指数运算时，法则为 (p/q)⁻ⁿ = (q/p)ⁿ。本质上就是先将这个分数上下颠倒（取倒数），然后再进行 n 次正乘方。这是IGCSE考试中非常常见的化简技巧。

For example: (2/3)⁻² = (3/2)² = 9/4 (5/1)⁻³ = (1/5)³ = 1/125 (1/4)⁻¹ = 4¹ = 4 This rule works because (a/b)⁻ⁿ = 1 / (a/b)ⁿ = 1 / (aⁿ/bⁿ) = bⁿ/aⁿ = (b/a)ⁿ.

例如： (2/3)⁻² = (3/2)² = 9/4 (5/1)⁻³ = (1/5)³ = 1/125 (1/4)⁻¹ = 4¹ = 4 这个法则成立是因为 (a/b)⁻ⁿ = 1 / (a/b)ⁿ = 1 / (aⁿ/bⁿ) = bⁿ/aⁿ = (b/a)ⁿ。

4. Combining with Multiplication and Division | 结合乘除运算

× a⁻ⁿ = aᵐ⁺⁽⁻ⁿ⁾ = aᵐ⁻ⁿ. When dividing, subtract the exponents: aᵐ ÷ a⁻ⁿ = aᵐ⁻⁽⁻ⁿ⁾ = aᵐ⁺ⁿ.

负指数常常与其他指数律同时出现。当同底数的幂相乘时，指数相加：aᵐ × a⁻ⁿ = aᵐ⁺⁽⁻ⁿ⁾ = aᵐ⁻ⁿ。当同底数的幂相除时，指数相减：aᵐ ÷ a⁻ⁿ = aᵐ⁻⁽⁻ⁿ⁾ = aᵐ⁺ⁿ。

Example: Simplify x³ × x⁻⁵. x⁻² = 1/x². The final answer is usually given with positive indices, so 1/x² is preferred.

示例：化简 x³ × x⁻⁵。 x³ × x⁻⁵ = x³⁻⁵ = x⁻² = 1/x²。最终答案通常需要表示为正指数形式，因此 1/x² 更合适。

t in the denominator effectively moves the term to the numerator with a positive exponent.

对于除法：(y⁴)/(y⁻²) = y⁴⁻⁽⁻²⁾ = y⁴⁺² = y⁶。分母中的负指数实际上相当于把该项移到分子并变成正指数。

5. Raising Powers to Negative Powers | 幂的负指数

Again, you would often rewrite the final expression with positive indices: a⁻ᵐⁿ = 1 / aᵐⁿ.

当一个幂再次进行乘方且指数为负时，指数相乘：(aᵐ)⁻ⁿ = aᵐ × ⁽⁻ⁿ⁾ = a⁻ᵐⁿ。同样，通常需要将最终结果改写为正指数：a⁻ᵐⁿ = 1 / aᵐⁿ。

Example: (z²)⁻³ = z² × ⁽⁻³⁾ = z⁻⁶ = 1/z⁶. hereas 2x⁻² means 2 × (1/x²) = 2/x². The placement of the bracket makes a huge difference.

示例：(z²)⁻³ = z² × ⁽⁻³⁾ = z⁻⁶ = 1/z⁶。 注意括号的使用：(2x)⁻² 表示 1/(2x)² = 1/(4x²)，而 2x⁻² 表示 2 × (1/x²) = 2/x²。括号的位置会造成巨大差异。

6. Negative Indices in Algebraic Expressions | 代数式中的负指数

In algebraic simplification, you are often expected to express terms with negative indices as fractions and then combine them. For example, to simplify (3a⁻²b³) / (a⁵b⁻¹), you can treat each variable separately using the division law, or rewrite negative exponents as reciprocals first.

在代数化简中，通常要求把含有负指数的项写成分数形式，然后再合并。例如，化简 (3a⁻²b³) / (a⁵b⁻¹)，你可以利用除法法则分别处理每个变量，或者先将负指数改写为倒数形式。

Using the laws of indices: 3a⁻²b³ ÷ a⁵b⁻¹ = 3a⁻²⁻⁵ b³⁻⁽⁻¹⁾ = 3a⁻⁷ b⁴ = (3b⁴) / a⁷. Always present the final answer with positive indices.

运用指数运算律： 3a⁻²b³ ÷ a⁵b⁻¹ = 3a⁻²⁻⁵ b³⁻⁽⁻¹⁾ = 3a⁻⁷ b⁴ = (3b⁴) / a⁷。 最终答案必须保留正指数。

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¹ + y⁻¹ = 1/x + 1/y = (y+x)/xy, so the whole expression becomes 1 / ((x+y)/xy) = xy/(x+y).

更复杂的表达式，例如 (x⁻¹ + y⁻¹)⁻¹，需要谨慎处理。和的倒数不等于倒数的和；你必须先合括号内的项：x⁻¹ + y⁻¹ = 1/x + 1/y = (y+x)/xy，因此整个表达式变为 1 / ((x+y)/xy) = xy/(x+y)。

7. Common Mistakes to Avoid | 常见错误

1/9. The negative sign in front of the base without brackets acts as a coefficient of −1.

一个常见错误是将指数前的负号与底数的符号混淆。例如，−3⁻² 不等于 (−3)⁻²。−3⁻² 表示 −(1/3²) = −1/9，而 (−3)⁻² = 1/(−3)² = 1/9。没有括号直接写在底数前的负号，其作用相当于一个系数 −1。

ds to messy results. The cleaner approach is to flip the fraction first: (a/b)⁻ⁿ = (b/a)ⁿ.

另一个错误是对分数负指数法则的错误应用：有些学生错误地写成 (a/b)⁻ⁿ = a⁻ⁿ / b⁻ⁿ，这数学上没有问题，但往往会导致混乱。更简洁的方法是先翻转分数：(a/b)⁻ⁿ = (b/a)ⁿ。

Also, never leave a negative exponent in a final simplified answer unless the question specifically asks you to retain index form. Always rewrite as a positive exponent with a denominator.

此外，除非题目明确要求保留指数形式，否则切勿在最终简化答案中留下负指数。始终将其改写为分母含有正指数的形式。

8. Solving Equations with Negative Indices | 解含负指数的方程

require rewriting the terms so that both sides share the same base. For example, solve 2⁻ˣ = 1/8. Recognizing that 1/8 = 8⁻¹ = (2³)⁻¹ = 2⁻³, we set 2⁻ˣ = 2⁻³, therefore −x = −3, giving x = 3.

涉及负指数的方程通常需要将各项改写为同底数的幂。例如，解方程 2⁻ˣ = 1/8。注意到 1/8 = 8⁻¹ = (2³)⁻¹ = 2⁻³，我们得到 2⁻ˣ = 2⁻³，因此 −x = −3，解得 x = 3。

ite as x⁻² = 25. Then take the reciprocal of the power: (x⁻²)⁻½ = 25⁻½, so x = (25)⁻½ = 1/√25 raising both sides to a negative fractional power, careful handling of signs is essential.

如果未知数出现在分母中，例如 1 / x² = 25，可以改写为 x⁻² = 25。然后对两边同时进行 −½ 次方（取倒数再开方）：(x⁻²)⁻½ = 25⁻½，得到 x = (25)⁻½ = 1/√25 = 1/5 或 −1/5。当两边进行负分数指数运算时，务必小心处理符号。

9. Harder Examples and Simplification | 难题与化简

Let' n: (4a⁻³b²)⁻² × (2a²b⁻¹)³. Start by applying the power to each factor inside the brackets.

我们来看一个多步化简的例子：(4a⁻³b²)⁻² × (2a²b⁻¹)³。首先将括号外的指数分别作用于括号内的每个因式。

First bracket: (4a⁻³b²)⁻² = 4⁻² × a⁽⁻³⁾⁽⁻²⁾ × b²⁽⁻²⁾ = (1/16) × a⁶ × b⁻⁴. Second bracket: (2a²b⁻¹)³ = 2³ × a⁶ × b⁻³ = 8a⁶b⁻³. Now multiply: (1/16) a⁶ b⁻⁴ × 8a⁶ b⁻³ = (8/16) a⁶⁺⁶ b⁻⁴⁻³ = (1/2) a¹² b⁻⁷ = a¹² / (2b⁷).

第一个括号：(4a⁻³b²)⁻² = 4⁻² × a⁽⁻³⁾⁽⁻²⁾ × b²⁽⁻²⁾ = (1/16) × a⁶ × b⁻⁴。 第二个括号：(2a²b⁻¹)³ = 2³ × a⁶ × b⁻³ = 8a⁶b⁻³。 现在相乘：(1/16) a⁶ b⁻⁴ × 8a⁶ b⁻³ = (8/16) a⁶⁺⁶ b⁻⁴⁻³ = (1/2) a¹² b⁻⁷ = a¹² / (2b⁷)。

se division by x⁻¹ means multiplying by x¹. Then combine: 3x⁻²⁺¹ y¹⁺³ = 3x⁻¹ y⁴ = 3y⁴ / x.

另一个典型问题：不用负指数表示 (3x⁻²y) / (x⁻¹y⁻³)。将其改写为 3x⁻²y × x¹y³，因为除以 x⁻¹ 相当于乘以 x¹。然后合并：3x⁻²⁺¹ y¹⁺³ = 3x⁻¹ y⁴ = 3y⁴ / x。

10. Summary and Key Points | 总结与要点

l the standard laws of indices apply, but you must be comfortable converting between negative and positive exponent forms to achieve the cleanest final answer.

负指数只是倒数的一种简洁记法。核心法则是对于任何非零数 a，有 a⁻ⁿ = 1 / aⁿ。所有标准指数运算律都适用，但你必须能灵活地在负指数与正指数形式之间转换，以便得到最简明的最终答案。

• Key points to remember:

  记忆要点：

• a⁻ⁿ = 1/aⁿ; 1/a⁻ⁿ = aⁿ

• (p/q)⁻ⁿ = (q/p)ⁿ

• When multiplying, add exponents; when dividing, subtract; when raising a power, multiply.

  相乘时指数相加；相除时指数相减；幂的乘方指数相乘。

• Always express your final answer with positive indices unless told otherwise.

  除非另有说明，最终答案一定要用正指数表示。

• Watch out for brackets and signs: (−a)⁻ⁿ vs −a⁻ⁿ can lead to completely different results.

  注意括号和符号：(−a)⁻ⁿ 与 −a⁻ⁿ 可能导致完全不同的结果。

Consistent practice with negative indices will make simplifying complex expressions and solving exponential equations much easier, building a strong foundation for topics like surds, logarithms, and advanced algebra.

通过持续练习负指数，你会发现化简复杂表达式以及求解指数方程变得轻松许多，这也会为根式、对数以及高级代数等主题打下坚实的基础。

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