📚 125: Perfect Cubes, Powers and Number Properties | 125：完全立方、幂与数的性质
The number 125 may look small, but it carries a surprisingly rich set of mathematical ideas. From being a perfect cube to showing up in standard form, percentages, and geometry, 125 is an excellent example for revising several IGCSE number topics.
数字125看起来不大，但它蕴含了非常丰富的数学思想。它既是完全立方，又出现在标准式、百分比和几何之中，是复习IGCSE数字类考点的绝佳例子。
1. What Makes 125 Special | 125的特殊之处
125 is a natural number that appears in many everyday and mathematical contexts. It is the result of multiplying 5 by itself three times: 5 × 5 × 5.
125是一个自然数，在日常生活中和数学中经常出现。它是5自身相乘三次的结果：5 × 5 × 5。
Because 125 can be written as a whole number raised to the third power, it belongs to the family of perfect cubes. The first few perfect cubes are 1, 8, 27, 64, 125, 216, and so on.
因为125可以写成一个整数的三次幂，所以它属于完全立方数家族。前几个完全立方数依次是1、8、27、64、125、216等等。
Recognising perfect cubes quickly is an important skill for simplifying cube roots and solving cubic equations.
快速识别完全立方数是化简立方根和求解三次方程的重要技能。
2. Prime Factorisation of 125 | 125的质因数分解
Prime factorisation means writing a number as a product of its prime factors. For 125, we divide by 5 repeatedly because 125 is divisible by 5.
质因数分解是指将一个数写成其质因数相乘的形式。对于125，因为能被5整除，我们用5不断去除。
- 125 ÷ 5 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1
So the prime factorisation of 125 is 5 × 5 × 5, often written as 5³.
因此125的质因数分解为5 × 5 × 5，通常写成5³。
125 = 5³
This compact notation instantly tells us how many times 5 appears as a factor.
这种简洁的表示法立刻告诉我们5作为因数出现了几次。
3. Index Notation: 125 = 5³ | 指数表示法：125 = 5³
In index notation, the base is 5 and the index (also called exponent or power) is 3. The index tells us how many times the base is used as a factor.
在指数表示法中，底数是5，指数（也称幂或次数）是3。指数告诉我们底数作为因数使用多少次。
5³ = 5 × 5 × 5 = 125
Understanding index notation allows you to solve problems such as 5³, which means 5 multiplied by itself three times.
理解指数表示法能让你解决类似5³的问题，即5自乘三次。
It also helps with the laws of indices. For example, 5² × 5³ = 5⁽²⁺³⁾ = 5⁵ = 3125. But 5⁶ ÷ 5³ = 5⁽⁶⁻³⁾ = 5³ = 125.
它还有助于应用指数法则。例如，5² × 5³ = 5⁽²⁺³⁾ = 5⁵ = 3125。而5⁶ ÷ 5³ = 5⁽⁶⁻³⁾ = 5³ = 125。
4. Cube Roots | 立方根
The cube root of a number is the value that, when cubed, gives the original number. For 125, the cube root is 5 because 5 × 5 × 5 = 125.
一个数的立方根是指哪个数自乘三次得到原数。对于125，立方根是5，因为5 × 5 × 5 = 125。
∛125 = 5
When solving equations of the form x³ = 125, the solution is x = ∛125 = 5.
当解形如x³ = 125的方程时，解为 x = ∛125 = 5。
Be careful: if x is a real number and x³ = 125, then x = 5 only, because cube roots of positive numbers are positive. However, for equations like x² = 125, there are two solutions: x = ±5√5.
注意：若x为实数且x³ = 125，则x = 5，因为正数的立方根为正数。但对于x² = 125这样的方程，有两个解：x = ±5√5。
5. Square Roots and Surds | 平方根与根式
125 is not a perfect square: there is no integer that, when squared, equals 125. But we can simplify its square root using prime factorisation.
125不是完全平方数：没有整数平方后等于125。但我们可以利用质因数分解化简它的平方根。
Since 125 = 25 × 5 and 25 is a perfect square, we get:
因为125 = 25 × 5，且25是完全平方数，我们得到：
√125 = √(25 × 5) = √25 × √5 = 5√5
So the simplified surd form of √125 is 5√5. This is a common skill in IGCSE questions.
因此√125的简化根式为5√5。这是IGCSE考试中常见的技能。
You should also be able to approximate √125 ≈ 11.18, but in algebra keep the exact form 5√5.
你也应该能够估算√125 ≈ 11.18，但在代数中请保留精确形式5√5。
6. Standard Form | 标准式
Standard form (also called scientific notation) is a way of writing numbers as a number between 1 and 10 multiplied by a power of 10.
标准式（也称科学计数法）是将一个数写成1到10之间的数乘以10的幂的形式。
To write 125 in standard form, move the decimal point two places left to get 1.25. Because 1.25 × 100 = 125, we use 10².
要把125写成标准式，将小数点向左移动两位得到1.25。因为1.25 × 100 = 125，所以我们用10²。
125 = 1.25 × 10²
This form is useful when dealing with very large or very small numbers, and it often appears in IGCSE calculators and real-life contexts such as distances in space.
这种形式在处理极大或极小的数字时非常有用，也常出现在IGCSE的计算器使用和现实情境中，例如太空距离。
7. Factors and Divisibility | 因数与整除性
A factor of a number is an integer that divides it exactly. Since 125 = 5³, its only prime factor is 5.
一个数的因数是指能够整除它的整数。因为125 = 5³，它的唯一质因数是5。
To list all positive factors of 125, consider powers of 5 from 5⁰ to 5³:
为了列出125的所有正因数，考虑从5⁰到5³的幂：
| Power of 5 | Value |
|---|---|
| 5⁰ | 1 |
| 5¹ | 5 |
| 5² | 25 |
| 5³ | 125 |
So the factors of 125 are 1, 5, 25, and 125. The number of factors is 4, which is (3 + 1) because the index in 5³ is 3.
因此125的因数为1、5、25和125。因数个数为4，即(3 + 1)，因为5³中的指数是3。
125 is also divisible by 5, 25, and 125 itself, but not by 2, 3, or 10.
125也能被5、25和125本身整除，但不能被2、3或10整除。
8. 125 in Geometry | 几何中的125
A cube of side length 5 units has a volume of 5 × 5 × 5 = 125 cubic units. Therefore, 125 is the volume of a 5 by 5 by 5 cube.
一个边长为5个单位的立方体，体积为5 × 5 × 5 = 125立方单位。因此，125是一个5×5×5立方体的体积。
Volume of cube = side³ = 5³ = 125
If you know the volume of a cube is 125 cm³, you can find its side length by taking the cube root: ∛125 = 5 cm.
若已知立方体体积为125 cm³，你可以通过开立方根求出边长：∛125 = 5 cm。
This connection between number and shape shows why perfect cubes are called “cubic” numbers.
这种数字与图形之间的联系说明了为什么完全立方数被称为“立方”数。
9. Percentages: 125% and Its Conversions | 百分比：125%及其换算
Percent means “out of 100”, so 125% = 125/100. As a fraction in lowest terms, this is 5/4, and as a decimal it is 1.25.
百分比的意思是“每一百”，所以125% = 125/100。化成最简分数为5/4，化成小数为1.25。
125% = 125/100 = 5/4 = 1.25
In percentage problems, a multiplier of 1.25 represents a 25% increase. For example, £100 increased by 25% becomes £100 × 1.25 = £125.
在百分比问题中，乘数1.25表示增加25%。例如，100英镑增加25%后变为100英镑 × 1.25 = 125英镑。
So 125 itself often appears as the result of applying a 25% increase to 100, or as a multiplier in reverse calculations.
所以125经常作为对100进行25%增加的结果出现，或者在反向计算中作为乘数出现。
10. Exam-Style Problem | 考试风格例题
Here is a typical IGCSE question that combines several ideas about 125.
下面是一道典型的IGCSE题目，综合了关于125的多个知识点。
- Solve x³ = 125
- Write √125 in the form a√5
- Express 125 in standard form
Solution: x³ = 125 gives x = ∛125 = 5. For √125, using prime factorisation, √125 = 5√5. For standard form, 125 = 1.25 × 10².
解答：x³ = 125 得 x = ∛125 = 5。对于√125，利用质因数分解得√125 = 5√5。对于标准式，125 = 1.25 × 10²。
These three steps test cube roots, surds, and standard form in one compact number.
这三个步骤通过同一个简洁的数字测试了立方根、根式和标准式。
11. Common Mistakes to Avoid | 常见错误提醒
A frequent error is confusing cube roots with square roots. Remember that ∛125 = 5, but √125 = 5√5, not 5.
一个常见错误是混淆立方根与平方根。记住∛125 = 5，但√125 = 5√5，而不是5。
Another mistake is thinking 125 is a prime number. It is not, because it has more than two factors: 1, 5, 25, 125.
另一个错误是认为125是质数。它不是质数，因为它有超过两个因数：1、5、25、125。
Also, be careful when using a calculator: 125^(1/3) equals 5, but 125^(1/2) equals approximately 11.18, not a whole number.
另外，使用计算器时注意：125^(1/3) 等于5，但125^(1/2) 约等于11.18，不是整数。
Finally, do not forget that standard form requires the first part to be between 1 and 10; 12.5 × 10¹ is not correct standard form.
最后，不要忘记标准式要求第一部分在1到10之间；12.5 × 10¹ 不是正确的标准式。
12. Summary | 总结
125 is a versatile number: it is 5³, its prime factorisation is 5³, its cube root is 5, and its square root simplifies to 5√5.
125是一个用途广泛的数字：它是5³，其质因数分解为5³，它的立方根是5，它的平方根化简为5√5。
It has exactly four positive factors, can be written as 1.25 × 10² in standard form, and appears naturally in geometry and percentages.
它恰好有四个正因数，可以写成标准式1.25 × 10²，并自然地出现在几何和百分比中。
Mastering these connections will help you solve IGCSE questions confidently and quickly.
掌握这些联系将帮助你在IGCSE考试中自信而快速地解题。
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