Standard Form & Indices | 标准形式与指数

📚 Standard Form & Indices | 标准形式与指数

Standard form (scientific notation) and the laws of indices are two of the most heavily tested topics in IGCSE Mathematics. They appear in almost every paper, whether as direct calculation questions or as tools inside questions on algebra, number, and problem solving. Mastering these skills will not only boost your marks but also make advanced topics such as logarithms and exponential functions much easier to understand later.

标准形式（科学记数法）与指数法则是IGCSE数学中考查频率最高的内容之一。几乎每张试卷都会出现，既可能作为直接计算题，也常作为代数、数字与实际问题求解的工具。掌握这些技能不仅能提高分数，还能让你今后学习对数与指数函数时更加轻松。

1. What Is Standard Form? | 什么是标准形式？

A number is written in standard form when it is expressed as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. The value A is called the coefficient or mantissa, and n is the power of ten. For example, 4,500 can be written as 4.5 × 10³, while 0.0045 can be written as 4.5 × 10⁻³.

当一个数表示为 A × 10ⁿ 的形式时，我们就说它使用了标准形式，其中 1 ≤ A < 10，n 为整数。A 称为系数或尾数，n 是十的幂次。例如，4,500 可以写成 4.5 × 10³，而 0.0045 可以写成 4.5 × 10⁻³。

The key rule is that the coefficient must be at least 1 but strictly less than 10. This is the most common source of lost marks: writing 25 × 10⁵ is incorrect in standard form, because 25 is not less than 10. The correct form is 2.5 × 10⁶.

关键规则是：系数必须大于或等于1且严格小于10。这是最常见的失分点：25 × 10⁵ 不是标准形式，因为25不小于10。正确写法是 2.5 × 10⁶。

Standard form is widely used in science to express very large or very small quantities, such as the distance from Earth to the Sun (1.5 × 10⁸ km) or the charge of an electron (1.6 × 10⁻¹⁹ C).

标准形式在科学中被广泛用于表示极大或极小的量，例如地球到太阳的距离（1.5 × 10⁸ 公里）或电子的电荷量（1.6 × 10⁻¹⁹ 库仑）。

2. Writing Large Numbers in Standard Form | 大数的标准形式写法

To convert a large positive number into standard form, place a decimal point after the first non-zero digit, then count how many digits follow it. This count becomes the positive index n.

要将一个较大的正数转换为标准形式，只需把小数点放在第一个非零数字之后，然后数一数其后还有多少位数，这个位数就是正指数 n。

Consider 78,000. The first non-zero digit is 7, so we write 7.8. The digits after 7 are 8, 0, 0, and 0: four digits in total. Therefore 78,000 = 7.8 × 10⁴.

以 78,000 为例。第一个非零数字是7，因此写成7.8。7后面的数字是8、0、0、0，共4位，所以 78,000 = 7.8 × 10⁴。

Here are more examples:

更多示例如下：

5,000 = 5 × 10³
23,700 = 2.37 × 10⁴
1,250,000 = 1.25 × 10⁶
600,000,000 = 6 × 10⁸

To convert back from standard form to an ordinary number, multiply the coefficient by the power of ten, which means moving the decimal point n places to the right. For example, 3.2 × 10⁵ = 320,000.

要从标准形式转换回普通数字，只需将系数乘以十的幂，也就是把小数点向右移动 n 位。例如，3.2 × 10⁵ = 320,000。

3. Writing Small Numbers in Standard Form | 小数的标准形式写法

For numbers less than 1, the process is similar, but the index is negative. Place the decimal point after the first non-zero digit, then count the number of places the decimal point has moved to the right. This gives the negative index −n.

对于小于1的数，方法类似，但指数为负。将小数点放在第一个非零数字之后，然后数一数小数点向右移动了多少位，得到的数就是负指数 −n。

Take 0.00092. The first non-zero digit is 9, so we write 9.2. The original decimal point has moved 4 places to the right, so the index is −4. Therefore 0.00092 = 9.2 × 10⁻⁴.

以 0.00092 为例。第一个非零数字是9，写成9.2。原小数点向右移动了4位，所以指数为 −4。因此 0.00092 = 9.2 × 10⁻⁴。

More examples:

更多示例：

0.05 = 5 × 10⁻²
0.000 007 2 = 7.2 × 10⁻⁶
0.000 000 45 = 4.5 × 10⁻⁷
0.000 1 = 1 × 10⁻⁴

To convert a negative-index standard form back to an ordinary number, move the decimal point n places to the left. For example, 6.5 × 10⁻⁵ = 0.000065.

要将负指数的标准形式转换回普通数字，把小数点向左移动 n 位。例如，6.5 × 10⁻⁵ = 0.000065。

4. The Laws of Indices | 指数运算法则

The laws of indices allow us to simplify expressions involving powers. These laws apply to any real base a, as long as division does not involve zero. They are essential for both algebra and standard form calculations.

指数法则帮助我们化简含幂的表达式。这些法则适用于任意实数底数 a（只要除法不涉及0即可）。它们是代数计算与标准形式运算的基础。

Index Law \| 指数法则	Example \| 示例
aᵐ × aⁿ = aᵐ⁺ⁿ	5² × 5³ = 5⁵ = 3125
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	10⁷ ÷ 10³ = 10⁴
(aᵐ)ⁿ = aᵐⁿ	(3²)⁴ = 3⁸
a⁰ = 1	7⁰ = 1
a⁻ᵐ = 1/aᵐ	2⁻³ = 1/8
a¹⁄ⁿ = ⁿ√a	9¹ᐟ² = √9 = 3
aᵐ⁄ⁿ = (ⁿ√a)ᵐ	8²ᐟ³ = (∛8)² = 4

When multiplying expressions with the same base, add the indices. When dividing, subtract the indices. When raising a power to another power, multiply the indices. A zero exponent always gives 1. A negative exponent creates a reciprocal, and a fractional exponent represents a root.

同底数幂相乘，指数相加；同底数幂相除，指数相减；幂的乘方，指数相乘。零次幂恒等于1。负指数表示取倒数，分数指数表示开方。

5. Negative and Fractional Indices | 负指数与分数指数

Negative indices do not make a number negative; they indicate a reciprocal. For example, 10⁻² means 1/10² = 1/100 = 0.01. In standard form, the negative index tells us that the original number is smaller than 1.

负指数并不表示负数，而是表示倒数。例如，10⁻² 表示 1/10² = 1/100 = 0.01。在标准形式中，负指数说明原数小于1。

Fractional indices represent roots. The expression x¹ᐟ² means the square root of x, and x¹ᐟ³ means the cube root of x. For example, 16¹ᐟ² = 4, and 27¹ᐟ³ = 3.

分数指数表示开方。x¹ᐟ² 表示 x 的平方根，x¹ᐟ³ 表示 x 的立方根。例如，16¹ᐟ² = 4，27¹ᐟ³ = 3。

More complex fractional indices combine a root and a power. For instance, 8²ᐟ³ means the cube root of 8, then squared: (∛8)² = 2² = 4. The order does not matter: (√16)³ = 4³ = 64 and √(16³) = √4096 = 64.

更复杂的分数指数结合了开方与乘方。例如，8²ᐟ³ 表示先对8开立方再平方：(∛8)² = 2² = 4。运算顺序不影响结果：(√16)³ = 4³ = 64，√(16³) = √4096 = 64。

A common exam question asks you to evaluate 5⁻² or (1/4)⁻¹. Remember that (1/4)⁻¹ = 4. In general, a negative exponent flips the fraction. Practise these two patterns until they feel automatic.

常见的考题要求计算 5⁻² 或 (1/4)⁻¹。注意 (1/4)⁻¹ = 4。一般来说，负指数会把分数倒过来。多加练习，直到这两种题型形成条件反射。

6. Multiplying and Dividing in Standard Form | 标准形式的乘除运算

To multiply two numbers in standard form, multiply the coefficients and add the powers of ten. Then, if the new coefficient is 10 or greater, rewrite it in proper standard form.

两个标准形式相乘时，系数相乘，十的幂相加。然后，如果新系数大于或等于10，需要重新改写成规范的标准形式。

(2 × 10⁵) × (3 × 10³) = 6 × 10⁸

Here 2 × 3 = 6, and 10⁵ × 10³ = 10⁸ since we add the exponents.

这里 2 × 3 = 6，且 10⁵ × 10³ = 10⁸，因为指数相加。

(4 × 10⁶) × (5 × 10²) = 20 × 10⁸ = 2 × 10⁹

Because 20 is not between 1 and 10, we rewrite 20 × 10⁸ as 2.0 × 10⁹, which means moving the decimal point one place and increasing the index by one.

因为20不在1到10之间，所以将 20 × 10⁸ 改写为 2.0 × 10⁹，即小数点左移一位、指数加一。

To divide, divide the coefficients and subtract the powers. For example:

除法运算中，系数相除，幂次相减。例如：

(6 × 10⁹) ÷ (2 × 10³) = 3 × 10⁶

(8 × 10⁴) ÷ (5 × 10⁻²) = 1.6 × 10⁶

In the second example, 8 ÷ 5 = 1.6, and 10⁴ ÷ 10⁻² = 10⁴⁻⁽⁻²⁾ = 10⁶. The result 1.6 × 10⁶ is already in standard form because 1.6 lies between 1 and 10.

第二个例子中，8 ÷ 5 = 1.6，且 10⁴ ÷ 10⁻² = 10⁴⁻⁽⁻²⁾ = 10⁶。结果 1.6 × 10⁶ 已经是标准形式，因为1.6在1到10之间。

7. Adding and Subtracting in Standard Form | 标准形式的加减运算

Addition and subtraction in standard form are different from multiplication and division: you cannot simply add or subtract the coefficients unless the powers of ten are exactly the same.

标准形式的加减法与乘除法不同：只有当两个数的十的幂完全相同时，才能直接对系数进行加减。

If the powers are the same, keep the power and

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Standard Form & Indices | 标准形式与指数