📚 Number 8: Standard Form and Indices | 数字第8讲：标准形式与指数
In Edexcel IGCSE Mathematics, the eighth core topic under 'Number' focuses on standard form and the laws of indices. Standard form gives us a compact way to write very large or very small numbers, while indices allow us to manipulate powers efficiently. This topic appears frequently in both calculator and non-calculator papers.
在Edexcel IGCSE数学中，“数字”部分的第八个核心主题聚焦于标准形式和指数法则。标准形式为我们提供了一种紧凑地书写极大或极小数字的方法，而指数则让我们能够高效地处理幂运算。该主题在计算器和非计算器试卷中都频繁出现。
1. What Is Standard Form? | 什么是标准形式？
A number is in standard form when it is written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. The value of a must always be at least 1 but less than 10, and n tells us how many places the decimal point has moved.
当一个数写成a × 10ⁿ的形式时，它就是标准形式，其中1 ≤ a < 10，且n是整数。a的值必须始终大于等于1且小于10，而n告诉我们小数点移动了多少位。
For example, the mass of the Earth is about 5,970,000,000,000,000,000,000,000 kg. In standard form this is written as 5.97 × 10²⁴ kg. The mass of an electron is about 0.000000000000000000000000000000911 kg, which is written as 9.11 × 10⁻³¹ kg.
例如，地球的质量约为5,970,000,000,000,000,000,000,000千克。用标准形式写为5.97 × 10²⁴千克。电子的质量约为0.000000000000000000000000000000911千克，可写为9.11 × 10⁻³¹千克。
2. The Laws of Indices | 指数法则
Indices, also called powers or exponents, follow a set of rules that allow us to simplify expressions involving powers. These laws are essential for working with standard form and many other algebra topics.
指数，也称为幂或次方，遵循一套规则，使我们可以简化涉及幂的表达式。这些法则是处理标准形式以及许多其他代数主题的基础。
Rule 1: When multiplying powers with the same base, add the indices.
法则1：同底数幂相乘时，指数相加。
aᵐ × aⁿ = aᵐ⁺ⁿ
Rule 2: When dividing powers with the same base, subtract the indices.
法则2：同底数幂相除时，指数相减。
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Rule 3: When raising a power to another power, multiply the indices.
法则3：幂的幂时，指数相乘。
(aᵐ)ⁿ = aᵐⁿ
Rule 4: Any non-zero number raised to the power 0 equals 1.
法则4：任何非零数的0次幂等于1。
a⁰ = 1
Rule 5: A negative index represents the reciprocal of the positive power.
法则5：负指数表示相应正指数的倒数。
a⁻ⁿ = 1 ÷ aⁿ
Rule 6: A fractional index represents a root. For example, a¹ᐟ² = √a and a¹ᐟ³ = ∛a.
法则6：分数指数表示根式。例如，a¹ᐟ² = √a，a¹ᐟ³ = ∛a。
a¹ᐟⁿ = ⁿ√a
3. Converting Ordinary Numbers into Standard Form | 将普通数字转换为标准形式
To convert a very large number into standard form, move the decimal point to the left until only one non-zero digit remains on its left. Count the number of places moved; this becomes the positive exponent n.
要将一个非常大的数转换为标准形式，将小数点向左移动，直到左边只剩下一位非零数字。记录移动的位数，这个位数就是正指数n。
For example: 84,000,000 = 8.4 × 10⁷. We moved the decimal point 7 places to the left.
例如：84,000,000 = 8.4 × 10⁷。我们将小数点向左移动了7位。
To convert a very small number into standard form, move the decimal point to the right until the first non-zero digit is just to the left of the decimal point. Count the places moved; this becomes the negative exponent n.
要将一个非常小的数转换为标准形式，将小数点向右移动，直到第一个非零数字位于小数点的左边。记录移动的位数，这个位数就是负指数n。
For example: 0.000 032 = 3.2 × 10⁻⁵. We moved the decimal point 5 places to the right.
例如：0.000 032 = 3.2 × 10⁻⁵。我们将小数点向右移动了5位。
4. Converting Standard Form back into Ordinary Numbers | 将标准形式还原为普通数字
To convert a number in standard form back into an ordinary number, multiply a by 10ⁿ. If n is positive, move the decimal point n places to the right. If n is negative, move the decimal point |n| places to the left.
要将标准形式还原为普通数字，用a乘以10ⁿ。如果n为正，将小数点向右移动n位；如果n为负，将小数点向左移动|n|位。
For example: 2.5 × 10³ = 2500. The decimal point moves 3 places to the right.
例如：2.5 × 10³ = 2500。小数点向右移动3位。
For example: 7.1 × 10⁻⁴ = 0.00071. The decimal point moves 4 places to the left.
例如：7.1 × 10⁻⁴ = 0.00071。小数点向左移动4位。
5. Multiplying Numbers in Standard Form | 标准形式的乘法
ts together. Then adjust the result so that the new a-value is in the interval 1 ≤ a < 10.
两个标准形式数相乘时，将前面的a值相乘，并将指数相加。然后调整结果，使新的a值处于1 ≤ a < 10的区间内。
Example: Calculate (3 × 10⁵) × (2 × 10⁴).
例题：计算(3 × 10⁵) × (2 × 10⁴)。
First multiply: 3 × 2 = 6. Then add exponents: 10⁵ × 10⁴ = 10⁹. The answer is 6 × 10⁹, which is already in standard form.
先相乘：3 × 2 = 6。再指数相加：10⁵ × 10⁴ = 10⁹。答案为6 × 10⁹，已符合标准形式。
u must re-adjust. For instance, (5 × 10²) × (4 × 10³) = 20 × 10⁵, which becomes 2.0 × 10⁶.
如果a值的乘积大于等于10，则需要重新调整。例如，(5 × 10²) × (4 × 10³) = 20 × 10⁵，应调整为2.0 × 10⁶。
6. Dividing Numbers in Standard Form | 标准形式的除法
When dividing numbers in standard form, divide the a-values and subtract the exponents. Again, check that the final a-value is between 1 and 10.
标准形式数相除时，将a值相除，并减去指数。同样，检查最终的a值是否在1到10之间。
Example: Calculate (8 × 10⁷) ÷ (2 × 10³).
例题：计算(8 × 10⁷) ÷ (2 × 10³)。
Divide a-values: 8 ÷ 2 = 4. Subtract exponents: 10⁷ ÷ 10³ = 10⁴. The answer is 4 × 10⁴.
a值相除：8 ÷ 2 = 4。指数相减：10⁷ ÷ 10³ = 10⁴。答案为4 × 10⁴。
exponents when subtracting. For example, (6 × 10⁵) ÷ (3 × 10⁻²) = 2 × 10⁵⁻⁽⁻²⁾ = 2 × 10⁷.
当指数为负时做减法要小心。例如，(6 × 10⁵) ÷ (3 × 10⁻²) = 2 × 10⁵⁻⁽⁻²⁾ = 2 × 10⁷。
7. Adding and Subtracting Numbers in Standard Form | 标准形式的加法和减法
Addition and subtraction in standard form are only straightforward when the powers of 10 are the same. If they are different, convert one of the numbers so that both have the same exponent.
只有当10的幂相同时，标准形式的加法和减法才是直接的。如果不同，将其中一个数转换，使两个数的指数相同。
Example: Calculate (2 × 10⁶) + (3 × 10⁵).
例题：计算(2 × 10⁶) + (3 × 10⁵)。
Rewrite 3 × 10⁵ as 0.3 × 10⁶. Then add: 2 × 10⁶ + 0.3 × 10⁶ = 2.3 × 10⁶.
将3 × 10⁵改写为0.3 × 10⁶。然后相加：2 × 10⁶ + 0.3 × 10⁶ = 2.3 × 10⁶。
For subtraction, use the same process; align the exponents first, then subtract the a-values.
减法使用相同的方法：先对齐指数，再减去a值。
8. Using Standard Form on a Calculator | 在计算器上使用标准形式
Scientific calculators have an EXP or ×10ⁿ s in standard form directly. For example, to enter 4.2 × 10⁸, press 4.2, then EXP, then 8.
科学计算器上有EXP或×10ⁿ按键，可以直接输入标准形式的数。例如，输入4.2 × 10⁸，按4.2，然后按EXP，再按8。
When the calculator displays a result like 3.5E12, this means 3.5 × 10¹². The 'E' stands for exponent and represents '×10 to the power of'.
当计算器显示结果如3.5E12时，这表示3.5 × 10¹²。这里的“E”代表指数，表示“乘以10的幂”。
In an exam, always write your final answer in standard form unless the question tells you otherwise. This is especially important for very large or very small answers.
在考试中，除非题目另有要求，否则最终答案应以标准形式书写。这对非常大或非常小的答案尤为重要。
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Mistake 1: Writing the a-value outside the required range, such as 12 × 10⁶ instead of 1.2 × 10⁷.
错误1：将a值写在了规定范围之外，例如把12 × 10⁶写成，而正确应为1.2 × 10⁷。
Mistake 2: Confusing the sign of the exponent when converting small numbers. Remember: numbers less than 1 always have a negative exponent in standard form.
错误2：转换小数时混淆指数的正负号。记住：小于1的数在标准形式中指数始终为负。
Mistake 3: Adding exponents when multiplying numbers in ordinary form instead of in standard form. Only use the laws of indices when the bases are the same and the values are written as powers.
错误3：在普通数字乘法中错误地使用指数加法。只有底数相同且写成幂的形式时，才能使用指数法则。
Mistake 4: Forgetting to readjust after multiplication or division. Always check that the final a-value is between 1 and 10.
错误4：乘法或除法后忘记重新调整。始终检查最终的a值是否在1到10之间。
10. Worked Exam Examples | 考试例题精讲
Example 1: Write 0.000 056 in standard form.
例题1：将0.000 056写成标准形式。
Move the decimal point after the 5, which is 5 places to the right. Therefore, 0.000 056 = 5.6 × 10⁻⁵.
将小数点向右移动到数字5后面，共5位。因此，0.000 056 = 5.6 × 10⁻⁵。
Example 2: Calculate (1.5 × 10³) × (4 × 10⁻²), giving your answer in standard form.
例题2：计算(1.5 × 10³) × (4 × 10⁻²)，答案用标准形式表示。
Multiply a-values: 1.5 × 4 = 6. Add exponents: 10³ × 10⁻² = 10¹. The result is 6 × 10¹, which equals 60.
a值相乘：1.5 × 4 = 6。指数相加：10³ × 10⁻² = 10¹。结果是6 × 10¹，即60。
Example 3: Calculate (2.8 × 10⁵) + (3.2 × 10⁴), giving your answer in standard form.
例题3：计算(2.8 × 10⁵) + (3.2 × 10⁴)，答案用标准形式表示。
Rewrite 3.2 × 10⁴ as 0.32 × 10⁵. Then add: 2.8 × 10⁵ + 0.32 × 10⁵ = 3.12 × 10⁵.
将3.2 × 10⁴改写为0.32 × 10⁵。然后相加：2.8 × 10⁵ + 0.32 × 10⁵ = 3.12 × 10⁵。
11. Practice Questions | 练习题
Try these exercises on your own to build confidence. Write your answers in standard form where appropriate.
请独立完成以下练习以增强信心。请在适当之处用标准形式写出答案。
1. Write 3,450,000 in standard form.
1. 将3,450,000写成标准形式。
2. Write 7.2 × 10⁻⁶ as an ordinary number.
2. 将7.2 × 10⁻⁶写成普通数字。
3. Calculate (4 × 10⁷) × (2 × 10⁻³).
3. 计算(4 × 10⁷) × (2 × 10⁻³)。
4. Calculate (9 × 10⁸) ÷ (3 × 10⁵).
4. 计算(9 × 10⁸) ÷ (3 × 10⁵)。
5. Calculate (1.2 × 10⁴) + (8.5 × 10³).
5. 计算(1.2 × 10⁴) + (8.5 × 10³)。
Answers: 1) 3.45 × 10⁶ 2) 0.0000072 3) 8 × 10⁴ 4) 3 × 10³ 5) 2.05 × 10⁴
答案：1) 3.45 × 10⁶ 2) 0.0000072 3) 8 × 10⁴ 4) 3 × 10³ 5) 2.05 × 10⁴
12. Summary | 总结
Standard form is a powerful tool for representing extreme numbers in science and finance. The key skills are: converting between standard form and ordinary notation, applying the laws of indices, and performing arithmetic operations while keeping the a-value between 1 and 10.
标准形式是表示科学和金融中极端数字的有力工具。关键技能包括：在标准形式和普通记数法之间转换、应用指数法则，以及在保持a值介于1到10之间的前提下进行算术运算。
Mastery of this topic comes from regular practice. Remember to check your answers on the calculator, but also learn to do simple conversions without a calculator, as non-calculator questions are common in the Edexcel IGCSE exam.
掌握这一主题需要经常练习。记得用计算器检查答案，但同时也要学会不用计算器进行简单转换，因为在Edexcel IGCSE考试中，非计算器题目很常见。
Now that you have reviewed 'Number 8', try applying these skills to past paper questions to reinforce your understanding.
现在你已经复习了“数字第8讲”，尝试将这些技能
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