📚 Powers of 10 and Standard Form | 10 的幂与标准形式
In IGCSE Mathematics, powers of 10 and standard form appear in both calculator and non-calculator papers. This topic is tested across number, algebra, ratio, and problem-solving questions. It is also essential for science subjects, where very large and very small quantities are common. This revision guide will help you understand the rules, avoid common mistakes, and answer exam questions with confidence.
在 IGCSE 数学中，10 的幂与标准形式在计算器试卷和非计算器试卷中都会出现。这个主题贯穿数、代数、比例和应用题，同时也是科学科目中处理极大或极小数值的重要基础。本复习指南将帮助你理解规则、避开常见错误，并自信地解答考试题目。
1. Index Notation | 指数记号
Index notation is a compact way of writing repeated multiplication. In the expression 2⁵, the base is 2 and the exponent or index is 5. This means 2 × 2 × 2 × 2 × 2 = 32. The word ‘power’ is also used, for example, 2⁵ is read as ‘two to the power 5’.
指数记号是一种简写重复乘法的方式。在表达式 2⁵ 中，底数是 2，指数是 5。它的意思是 2 × 2 × 2 × 2 × 2 = 32。我们还会用到 “幂” 这个词，例如 2⁵ 读作 “2 的 5 次幂”。
For powers of 10, the pattern is simple: 10¹ = 10, 10² = 100, 10³ = 1000, and so on. In particular, 10⁶ = 1,000,000, which is one million. The exponent shows how many zeros follow the 1 when the exponent is positive.
对于 10 的幂，规律非常简单：10¹ = 10，10² = 100，10³ = 1000，以此类推。特别地，10⁶ = 1,000,000，也就是一百万。当指数为正时，指数表示 1 后面有多少个 0。
10⁶ = 1,000,000
You must be able to write powers in expanded form and recognise powers when they appear inside a larger calculation. For example, 10⁴ × 10³ can be simplified before you work out the final value.
你必须能将幂展开为普通形式，也要能在更大计算中认出幂。例如，10⁴ × 10³ 可以先化简，再计算最终数值。
2. Rules of Indices | 指数法则
The rules of indices allow you to simplify expressions with powers quickly. These rules work when the bases are the same. The first rule says: when multiplying powers with the same base, add the exponents.
指数法则可以帮助你快速化简含幂的表达式。这些法则要求底数相同。第一条法则是：同底数幂相乘时，指数相加。
| Rule | Formula | Example |
| Multiplying | aᵐ × aⁿ = aᵐ⁺ⁿ | 3² × 3⁴ = 3⁶ = 729 |
| Dividing | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁸ ÷ 5³ = 5⁵ = 3125 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ = 64 |
These rules are used constantly in standard form calculations. For example, because 10⁶ means 10 raised to the power 6, multiplying 10⁶ by 10² gives 10⁸. Do not multiply the exponents when multiplying the bases.
这些法则在标准形式计算中被反复使用。例如，因为 10⁶ 表示 10 的 6 次方，所以 10⁶ × 10² = 10⁸。请记住：同底数幂相乘时，不要将指数相乘。
3. Zero, Negative and Fractional Powers | 零指数、负指数和分数指数
Any non-zero number raised to the power 0 equals 1. This may seem strange, but it follows from the division rule: a³ ÷ a³ = a⁰ and also equals 1. So 7⁰ = 1, 10⁰ = 1, and 999⁰ = 1.
任何非零数的 0 次幂都等于 1。这看起来有点奇怪，但它可以从除法法则推出：a³ ÷ a³ = a⁰，同时结果也等于 1。因此 7⁰ = 1，10⁰ = 1，999⁰ = 1。
A negative power means the reciprocal. For example, 10⁻¹ = 1 ÷ 10 = 0.1, and 10⁻³ = 1 ÷ 1000 = 0.001. Later you will see that negative powers are used for very small numbers in standard form.
负指数表示倒数。例如，10⁻¹ = 1 ÷ 10 = 0.1，10⁻³ = 1 ÷ 1000 = 0.001。稍后你会看到，标准形式中很小的数就用负指数来表示。
10⁻³ = 1 ÷ 10³ = 0.001
Fractional powers are connected to roots. The power 1/2 is the square root, the power 1/3 is the cube root, and the power 1/n is the nth root. For example, 9¹⁄₂ = √9 = 3 and 27¹⁄₃ = ∛27 = 3.
分数指数与根号有关。1/2 次幂是平方根，1/3 次幂是立方根，1/n 次幂是 n 次方根。例如，9¹⁄₂ = √9 = 3，27¹⁄₃ = ∛27 = 3。
4. What Is Standard Form? | 什么是标准形式
Standard form is a way of writing very large or very small numbers. A number is in standard form when it is written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. The number A is often called the mantissa or coefficient.
标准形式是书写很大或很小数字的一种方法。一个数写成 A × 10ⁿ 的形式就称为标准形式，其中 1 ≤ A < 10，n 是整数。A 常被称为尾数或系数。
If n is positive, the number is large. If n is negative, the number is small. For example, 4,500 can be written as 4.5 × 10³ because 4.5 × 1000 = 4500. The small number 0.00067 can be written as 6.7 × 10⁻⁴.
如果 n 是正数，表示这个数很大；如果 n 是负数，表示这个数很小。例如，4,500 可以写成 4.5 × 10³，因为 4.5 × 1000 = 4500。很小的数 0.00067 可以写成 6.7 × 10⁻⁴。
Standard form is sometimes called scientific notation. In Edexcel IGCSE exams, you must give final answers in standard form when the question asks for it, and you must keep the mantissa between 1 and 10.
标准形式有时也叫科学记数法。在 Edexcel IGCSE 考试中，如果题目要求使用标准形式，你必须按标准形式给出最终答案，并且尾数必须保持在 1 到 10 之间。
5. Converting Large Numbers | 将大数写成标准形式
To convert a large ordinary number into standard form, place the decimal point after the first non-zero digit. Then count how many places the decimal point has moved. This count becomes the positive exponent of 10.
要把一个较大的普通数转换为标准形式，先把小数点放在第一个非零数字之后。然后数一数小数点移动了多少位，这个位数就是 10 的正指数。
Example: convert 106,000 to standard form. The first non-zero digit is 1, so write 1.06. The decimal point moves 5 places to the left, so 106,000 = 1.06 × 10⁵.
例如：将 106,000 转换为标准形式。第一个非零数字是 1，所以写成 1.06。小数点向左移动了 5 位，因此 106,000 = 1.06 × 10⁵。
106,000 = 1.06 × 10⁵
To convert back, move the decimal point to the right. For example, 2.07 × 10⁶ = 2.07 × 1,000,000 = 2,070,000. The exponent tells you the number of places to move the decimal point.
反向转换时，把小数点向右移动。例如，2.07 × 10⁶ = 2.07 × 1,000,000 = 2,070,000。指数告诉你小数点需要移动多少位。
6. Converting Small Numbers | 将小数写成标准形式Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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