Indices and Standard Form | 指数与科学记数法

📚 Indices and Standard Form | 指数与科学记数法

Indices (or powers) and standard form are fundamental skills in IGCSE Mathematics. They allow us to write very large and very small numbers in a compact way, and they form the basis for many higher-level topics such as algebra, calculus and applied mathematics.

指数（或幂）与科学记数法是IGCSE数学中的基础技能。它们使我们能够以紧凑的方式书写非常大和非常小的数字，并且是代数、微积分和应用数学等许多高级主题的基础。

1. Index Notation and Bases | 指数记法与其底数

In the expression aⁿ, ‘a’ is called the base and ‘n’ is called the index (or exponent or power). It means that a is multiplied by itself n times. For example, 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.

在表达式aⁿ中，’a’称为底数，’n’称为指数（或幂）。它表示a自乘n次。例如，2⁵ = 2 × 2 × 2 × 2 × 2 = 32。

The plural of index is indices. When the index is 1, we usually just write the base, so a¹ = a. When the index is 0, we will see later that a⁰ = 1 (provided a ≠ 0). An index can be a positive integer, zero, a negative integer, or even a fraction.

index的复数是indices。当指数为1时，我们通常只写底数，所以a¹ = a。当指数为0时，稍后我们会看到a⁰ = 1（前提是a ≠ 0）。指数可以是正整数、零、负整数，甚至是分数。

2. First Law: Multiplying Powers | 法则一：同底数幂相乘

When we multiply two powers with the same base, we keep the base and add the indices. This is written as:

当我们把两个同底数的幂相乘时，我们保留底数并相加指数。可以写为：

aᵐ × aⁿ = aᵐ⁺ⁿ

For example, 3⁴ × 3² = 3⁴⁺² = 3⁶ = 729. Notice that 3⁴ = 81 and 3² = 9, so 81 × 9 = 729, which confirms the rule.

例如，3⁴ × 3² = 3⁴⁺² = 3⁶ = 729。注意3⁴ = 81，3² = 9，所以81 × 9 = 729，这验证了法则。

It is important to remember that this law only applies when the bases are the same. For example, 2³ × 3² cannot be simplified in this way.

需要记住，这条法则只在底数相同时才适用。例如，2³ × 3²不能这样化简。

3. Second Law: Dividing Powers | 法则二：同底数幂相除

When we divide two powers with the same base, we keep the base and subtract the indices:

当我们把两个同底数的幂相除时，我们保留底数并相减指数：

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

For example, 7⁶ ÷ 7⁴ = 7⁶⁻⁴ = 7² = 49. We can also write this as a fraction: 7⁶/7⁴ = 7².

例如，7⁶ ÷ 7⁴ = 7⁶⁻⁴ = 7² = 49。我们也可以把它写成分数形式：7⁶/7⁴ = 7²。

If m < n, the result will have a negative index. For example, 5³ ÷ 5⁵ = 5³⁻⁵ = 5⁻². We will study negative indices in Section 5.

如果m < n，结果将带有负指数。例如，5³ ÷ 5⁵ = 5³⁻⁵ = 5⁻²。我们将在第5节学习负指数。

4. Third Law: Power of a Power | 法则三：幂的乘方

When we raise a power to another power, we multiply the indices:

当我们把一个幂再乘方时，我们要相乘指数：

(aᵐ)ⁿ = aᵐⁿ

For example, (2³)⁴ = 2³×⁴ = 2¹² = 4096. This rule is especially useful when simplifying expressions with brackets.

例如，(2³)⁴ = 2³×⁴ = 2¹² = 4096。这条法则在化简带括号的表达式时特别有用。

We must be careful: (aᵐ)ⁿ is not the same as aᵐⁿ without brackets. The bracket changes the order of operations. For example, (2³)² = 8² = 64, while 2³² = 2⁹ = 512. They are completely different.

我们必须小心：(aᵐ)ⁿ与没有括号的aᵐⁿ不同。括号改变了运算顺序。例如，(2³)² = 8² = 64，而2³² = 2⁹ = 512。它们完全不同。

5. Zero and Negative Indices | 零指数与负指数

Using the division law, we can discover the meaning of zero and negative indices. Since aⁿ ÷ aⁿ = 1, but also aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, we get:

利用除法法则，我们可以发现零指数和负指数的含义。因为aⁿ ÷ aⁿ = 1，而同时aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰，所以我们得到：

a⁰ = 1 (a ≠ 0)

Similarly, a⁰ ÷ aⁿ = 1 ÷ aⁿ = a⁰⁻ⁿ = a⁻ⁿ. Therefore:

类似地，a⁰ ÷ aⁿ = 1 ÷ aⁿ = a⁰⁻ⁿ = a⁻ⁿ。因此：

a⁻ⁿ = 1/aⁿ (a ≠ 0)

For example, 3⁻² = 1/3² = 1/9. Negative indices indicate reciprocals, not negative numbers.

例如，3⁻² = 1/3² = 1/9。负指数表示倒数，而负数指数本身不是负数。

In exam questions, always convert negative indices to fractions before simplifying further. For instance, 2x⁻³ = 2/x³.

在考试题目中，总是先把负指数转换为分数再进一步化简。例如，2x⁻³ = 2/x³。

6. Fractional Indices | 分数指数

Fractional indices represent roots. The rule is:

分数指数表示根式。规则是：

a^(1/n) = ⁿ√a

For example, 16^(1/2) = √16 = 4, and 8^(1/3) = ³√8 = 2.

例如，16^(1/2) = √16 = 4，而8^(1/3) = ³√8 = 2。

For a general fractional index m/n, we write a^(m/n) = (ⁿ√a)ᵐ, or equivalently ⁿ√(aᵐ). This means we either take the root first and then raise to the power, or raise to the power first and then take the root; the result is the same.

对于一般的分数指数m/n，我们写 a^(m/n) = (ⁿ√a)ᵐ，或者等价地写成 ⁿ√(aᵐ)。这意味着我们可以先开方再乘方，或者先乘方再开方，结果是一样的。

a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)

For example, 27^(2/3) = (³√27)² = 3² = 9. Also, 9^(3/2) = (√9)³ = 3³ = 27.

例如，27^(2/3) = (³√27)² = 3² = 9。同样，9^(3/2) = (√9)³ = 3³ = 27。

7. Simplifying Expressions with Indices | 化简含有指数的表达式

When simplifying algebraic expressions with indices, we apply the laws together. For example:

化简含有指数的代数表达式时，我们综合应用这些法则。例如：

Simplify (2x³y⁻²)⁴ ÷ (x²y⁵)⁻¹

First, apply the power of a product and power of a power: (2x³y⁻²)⁴ = 2⁴ x¹² y⁻⁸. And (x²y⁵)⁻¹ = x⁻² y⁻⁵.

首先，应用积的乘方和幂的乘方：(2x³y⁻²)⁴ = 2⁴ x¹² y⁻⁸。而(x²y⁵)⁻¹ = x⁻² y⁻⁵。

Now divide: (2⁴ x¹² y⁻⁸) ÷ (x⁻² y⁻⁵) = 2⁴ x¹²⁻⁽⁻²⁾ y⁻⁸⁻⁽⁻⁵⁾ = 16 x¹⁴ y⁻³.

现在相除：(2⁴ x¹² y⁻⁸) ÷ (x⁻² y⁻⁵) = 2⁴ x¹²⁻⁽⁻²⁾ y⁻⁸⁻⁽⁻⁵⁾ = 16 x¹⁴ y⁻³。

The final answer can be written with a positive index as 16x¹⁴ / y³.

最终答案可以写成带有正指数的形式：16x¹⁴ / y³。

8. Standard Form: Writing Large and Small Numbers | 科学记数法：书写大数与小数

Standard form (or scientific notation) is a way of writing numbers as a product of a number between 1 and 10 and a power of 10. The general form is:

科学记数法（standard form）是一种将数字写成1到10之间的数与10的幂的乘积的形式。一般形式为：

A × 10ⁿ, where 1 ≤ A < 10 and n is an integer

For example, 5,600 = 5.6 × 10³. We move the decimal point 3 places to the left, so the index is 3.

例如，5,600 = 5.6 × 10³。我们向左移动小数点3位，所以指数为3。

For small numbers, we use negative indices. For example, 0.00047 = 4.7 × 10⁻⁴. We move the decimal point 4 places to the right, so the index is -4.

对于小数，我们使用负指数。例如，0.00047 = 4.7 × 10⁻⁴。我们向右移动小数点4位，所以指数为-4。

To convert from standard form to ordinary form, multiply the decimal part by the power of 10. For instance, 6.2 × 10⁵ = 620,000.

要从科学记数法转换为普通数，只需将小数部分乘以10的幂。例如，6.2 × 10⁵ = 620,000。

9. Calculating with Standard Form | 科学记数法的计算

When multiplying or dividing numbers in standard form, we can handle the decimals and the powers of 10 separately.

当对科学记数法形式的数字进行乘除时，我们可以分别处理小数部分和10的幂。

For multiplication, (a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ. The result may need adjusting if a × b is not between 1 and 10.

对于乘法，(a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ。如果a × b不在1到10之间，结果可能需要调整。

For example, (2.5 × 10⁶) × (4 × 10⁻²) = 10 × 10⁴ = 1.0 × 10⁵.

例如，(2.5 × 10⁶) × (4 × 10⁻²) = 10 × 10⁴ = 1.0 × 10⁵。

For addition or subtraction, we must first make sure the powers of 10 are the same. For example, (3 × 10⁴) + (5 × 10³) = (3 × 10⁴) + (0.5 × 10⁴) = 3.5 × 10⁴.

对于加法或减法，我们必须首先确保10的幂相同。例如，(3 × 10⁴) + (5 × 10³) = (3 × 10⁴) + (0.5 × 10⁴) = 3.5 × 10⁴。

Remember to leave the final answer in standard form.

记住要把最终答案保留为科学记数法形式。

10. Solving Simple Exponential Equations | 解简单的指数方程

Some equations can be solved by rewriting both sides with the same base. If aˣ = aⁿ, then x = n (for a > 0 and a ≠ 1).

有些方程可以通过将两边写成相同底数的幂来求解。如果aˣ = aⁿ，那么x = n（其中a > 0且a ≠ 1）。

For example, solve 8ˣ = 2⁹. Since 8 = 2³, we write 8ˣ = (2³)ˣ = 2³ˣ. Therefore 2³ˣ = 2⁹, so 3x = 9, giving x = 3.

例如，解8ˣ = 2⁹。因为8 = 2³，我们写8ˣ = (2³)ˣ = 2³ˣ。因此2³ˣ = 2⁹，所以3x = 9，得到x = 3。

This method is very common in IGCSE papers. It tests your ability to recognise powers of 2, 3, 4, 5, 8, 9 and 10 easily.

这种方法在IGCSE试卷中非常常见。它测试你快速识别2、3、4、5、8、9和10的幂的能力。

11. Common Mistakes and Exam Tips | 常见错误与考试技巧

Mistake: Applying the multiplication law to different bases. Remember aᵐ × bⁿ cannot be simplified unless a = b.
错误：对不同的底数应用乘法法则。记住aᵐ × bⁿ在a = b之前无法化简。
Mistake: Forgetting that a⁰ = 1, not 0.
错误：忘记a⁰ = 1，而不是0。
Mistake: Writing 2x⁻¹ as -2x. The correct form is 2/x.
错误：把2x⁻¹写成-2x。正确形式是2/x。
Mistake: In standard form, using A outside the range 1 ≤ A < 10. For example, 45 × 10³ is not standard form; it should be 4.5 × 10⁴.
错误：在科学记数法中，A不在1 ≤ A < 10的范围内。例如，45 × 10³不是科学记数法，应为4.5 × 10⁴。
Tip: Always simplify indices step by step and show every step clearly in the exam.
技巧：始终一步一步地化简指数，并在考试中清晰展示每一步。
Tip: Learn the common powers by heart: 2¹⁰ = 1024, 3⁵ = 243, 5⁴ = 625, etc.
技巧：牢记常见幂：2¹⁰ = 1024, 3⁵ = 243, 5⁴ = 625等。

12. Worked Example and Practice | 例题与练习

Worked Example: Express 0.000625 in standard form and evaluate (2 × 10⁻³)³.

例题：将0.000625用科学记数法表示，并计算(2 × 10⁻³)³。

Solution: 0.000625 = 6.25 × 10⁻⁴. For the cube, (2 × 10⁻³)³ = 2³ × (10⁻³)³ = 8 × 10⁻⁹ = 8 × 10⁻⁹. That is already in standard form.

解答：0.000625 = 6.25 × 10⁻⁴。对于立方，(2 × 10⁻³)³ = 2³ × (10⁻³)³ = 8 × 10⁻⁹ = 8 × 10⁻⁹。这已经是科学记数法形式。

Practice: Try these yourself. Answers are below.

练习：请自行尝试以下题目。答案在下方。

Simplify 5⁹ × 5⁴. | 化简5⁹ × 5⁴。
Evaluate 4⁻³. | 计算4⁻³。
Write 3.8 × 10⁻² as an ordinary number. | 把3.8 × 10⁻²写成普通数。
Express 2³¹ ÷ 2²⁷ as a single power. | 把2³¹ ÷ 2²⁷表示为单一幂。
Solve 9ˣ = 27. | 解9ˣ = 27。

Answers: 5¹³; 1/64; 0.038; 2⁴; x = 3/2

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Indices and Standard Form | 指数与科学记数法