📚 Integers, Powers and Roots | 整数、幂与根
Understanding integers, powers and roots is a fundamental skill in Key Stage 3 Mathematics. It allows you to work confidently with negative numbers, multiples, factors, primes, squares, cubes and more complicated expressions. This guide will take you through each concept step by step, with clear explanations and examples.
理解整数、幂与根是 KS3 数学阶段的基本技能。它能帮助你自信地处理负数、倍数、因数、质数、平方数、立方数以及更复杂的表达式。本指南将通过清晰的解释与实例，逐步带你掌握每个概念。
1. Using Negative Numbers | 使用负数
Negative numbers are numbers less than zero. They are used to represent temperatures below freezing, depths below sea level, or losses in money. On a number line, they appear to the left of zero.
负数是小于零的数。它们常用于表示冰点以下的温度、海平面以下的深度或金钱的亏损。在数轴上，负数位于零的左侧。
When adding or subtracting with negative numbers, remember: adding a negative is the same as subtracting its positive value. For example, 5 + (-3) = 5 – 3 = 2. Subtracting a negative is the same as adding its positive value: 4 – (-6) = 4 + 6 = 10.
对负数进行加减运算时，请记住：加上一个负数等于减去它的绝对值。例如，5 + (-3) = 5 – 3 = 2。减去一个负数等于加上它的绝对值：4 – (-6) = 4 + 6 = 10。
Multiplying or dividing with negative numbers follows these rules: positive × positive = positive; negative × negative = positive; positive × negative = negative. The same applies for division.
负数相乘或相除遵循以下法则：正数乘以正数得正数；负数乘以负数得正数；正数乘以负数得负数。除法同理。
| Operation | Example |
|---|---|
| (−2) × (−3) | 6 |
| (−4) ÷ 2 | −2 |
| 6 + (−7) | −1 |
| (−3) − (−5) | 2 |
2. Multiples and Factors | 倍数与因数
A multiple of a number is the result of multiplying that number by an integer. For example, the first five multiples of 3 are 3, 6, 9, 12 and 15. Multiples are used when finding common denominators or working with times tables.
一个数的倍数是指这个数乘以任意整数后得到的结果。例如，3 的前五个倍数是 3、6、9、12 和 15。在寻找公分母或处理乘法表时，会用到倍数。
A factor is a whole number that divides exactly into another number, leaving no remainder. For instance, the factors of 12 are 1, 2, 3, 4, 6 and 12. Every number has at least two factors: 1 and itself.
因数是指能整除另一个数且没有余数的整数。例如，12 的因数有 1、2、3、4、6 和 12。每个数都至少有两个因数：1 和它本身。
Common factors are factors shared by two or more numbers. The highest common factor (HCF) is the largest of these. Common multiples are multiples shared by numbers; the lowest common multiple (LCM) is the smallest non‑zero multiple they share.
公因数是两个或多个数共有的因数，其中最大的一个称为最大公因数（HCF）。公倍数是这些数共有的倍数，其中最小的非零倍数称为最小公倍数（LCM）。
3. Prime Numbers and Prime Factors | 质数与质因数
A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. Examples include 2, 3, 5, 7, 11 and 13. The number 1 is not prime because it only has one factor.
质数是大于 1 且只有两个因数（1 和自身）的整数。例如 2、3、5、7、11 和 13。数字 1 不是质数，因为它只有一个因数。
Every composite number can be written as a product of prime numbers. This is called prime factorisation. A factor tree helps break a number down into its prime factors. For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
每个合数都可以写成质数的乘积，这称为质因数分解。可以使用因子树将一个数分解为它的质因数。例如，60 = 2 × 2 × 3 × 5 = 2² × 3 × 5。
When expressing a number in index form, we write repeated multiplication using powers. So 2 × 2 × 2 is written as 2³.
用指数形式表示时，我们将重复的乘法用幂的形式书写。例如 2 × 2 × 2 写作 2³。
4. Powers and Roots | 幂与根
A power (or index) tells you how many times a number is multiplied by itself. The base is the number being multiplied. In 5², the base is 5 and the index is 2, meaning 5 × 5 = 25.
幂（或指数）表示一个数自乘多少次。底数是被乘的数。在 5² 中，底数是 5，指数是 2，表示 5 × 5 = 25。
The opposite of squaring is taking the square root. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, √25 = 5 because 5² = 25. Square roots can be positive or negative, but the symbol √ denotes the principal (positive) root.
平方的逆运算是开平方根。一个数的平方根是乘以自身后得到原数的值。例如，√25 = 5，因为 5² = 25。平方根可以是正数或负数，但符号 √ 表示算术平方根（正根）。
Similarly, cubing a number means raising it to the power of 3. The cube root reverses this. The cube root of 27 is 3, written as ∛27 = 3, because 3³ = 27. Negative numbers can also have real cube roots, e.g. ∛(−8) = −2.
同样地，将一个数立方是指将它自乘三次。立方根是其逆运算。27 的立方根是 3，写作 ∛27 = 3，因为 3³ = 27。负数也有实数立方根，例如 ∛(−8) = −2。
5. Square Numbers and Square Roots | 平方数与平方根
Square numbers are the result when an integer is multiplied by itself. The first ten square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Recognising these instantly helps with mental arithmetic and algebra.
平方数是将一个整数乘以自身得到的结果。前十个平方数是 1、4、9、16、25、36、49、64、81 和 100。熟练识别这些数字有助于心算和代数运算。
When evaluating the square root of a larger number that is not a perfect square, you can use your knowledge of perfect squares to estimate. For instance, √50 lies between 7 (since 7² = 49) and 8 (since 8² = 64), so approximately 7.1.
当计算一个非完全平方数的平方根时，你可以利用对完全平方数的掌握进行估算。例如，√50 位于 7（7² = 49）和 8（8² = 64）之间，因此大约是 7.1。
Negative numbers do not have real square roots because a real number squared cannot be negative. However, zero has a square root of zero.
负数没有实数平方根，因为任何实数的平方都不会是负数。而零的平方根是零。
6. Cube Numbers and Cube Roots | 立方数与立方根
A cube number is the result of multiplying an integer by itself twice more, i.e. n × n × n. The first six cube numbers are 1, 8, 27, 64, 125 and 216. Like square numbers, cube numbers grow quickly and are useful in geometry for volumes.
立方数是将一个整数自乘两次得到的结果，即 n × n × n。前六个立方数是 1、8、27、64、125 和 216。与平方数一样，立方数增长很快，并在几何学中用于计算体积。
The cube root is the number that was cubed. The cube root of a negative number is negative, because a negative cubed yields a negative. For example, (−2)³ = −8, so the cube root of −8 is −2.
立方根是立方数的逆运算。负数的立方根也是负数，因为负数立方后仍为负数。例如，(−2)³ = −8，因此 −8 的立方根为 −2。
Cube roots of positive numbers are always positive, but cube roots of negative numbers are real, unlike square roots.
正数的立方根恒为正数，而与平方根不同的是，负数的立方根是实数。
7. Indices and Index Laws | 指数与指数法则
Indices (also called exponents or powers) follow specific rules that make calculations easier. The multiplication law: when multiplying powers with the same base, add the indices. For example, 2³ × 2⁴ = 2⁷.
指数（也称为幂或指数）遵循一系列特定法则，以便简化计算。乘法法则：同底数幂相乘，指数相加。例如，2³ × 2⁴ = 2⁷。
The division law: when dividing powers with the same base, subtract the indices. For instance, 5⁶ ÷ 5² = 5⁴. Remember that the base must be identical for this rule to apply.
除法法则：同底数幂相除，指数相减。例如，5⁶ ÷ 5² = 5⁴。请记住，只有底数相同时才能使用该法则。
The power of a power law: when raising a power to another power, multiply the indices: (3²)⁴ = 3⁸. Also, any number to the power of zero equals 1, e.g. 10⁰ = 1.
幂的幂法则：算一个幂的幂时，将指数相乘：(3²)⁴ = 3⁸。此外，任何非零数的 0 次幂都等于 1，例如 10⁰ = 1。
- Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ
- Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- Power of a power: (aᵐ)ⁿ = aᵐⁿ
- Zero exponent: a⁰ = 1 (if a ≠ 0)
8. Order of Operations (BIDMAS) | 运算顺序（BIDMAS）
When an expression involves multiple operations, you must follow the order of operations to get the correct answer. The acronym BIDMAS helps you remember: Brackets, Indices, Division, Multiplication, Addition, Subtraction.
当一个表达式包含多种运算时，必须遵循运算顺序才能得到正确答案。助记词 BIDMAS 可以帮助记忆：括号、指数、除法、乘法、加法、减法。
Division and multiplication have equal priority and are performed from left to right. The same applies to addition and subtraction. For example, in 8 + 2 × 3, multiply first: 2 × 3 = 6, then add: 8 + 6 = 14.
除法和乘法优先级相同，按从左到右的顺序计算。加法和减法也是如此。例如在 8 + 2 × 3 中，先算乘法：2 × 3 = 6，再算加法：8 + 6 = 14。
If brackets are present, do everything inside the brackets first, including any indices or division that may be inside. For instance, (3 + 4)² ÷ 7 = 7² ÷ 7 = 49 ÷ 7 = 7.
如果有括号，先计算括号内的所有运算，包括可能出现的指数或除法。例如，(3 + 4)² ÷ 7 = 7² ÷ 7 = 49 ÷ 7 = 7。
9. Estimating Square and Cube Roots | 估算平方根与立方根
When a calculator is not available, estimating roots is a valuable skill. You need to know the nearest perfect squares or cubes. To estimate √38, note that 6² = 36 and 7² = 49, so √38 is just above 6, maybe around 6.2.
在不使用计算器的情况下，估算平方根是一项很有用的技能。你需要知道最接近的完全平方数或立方数。例如估算 √38，由于 6² = 36，7² = 49，因此 √38 略大于 6，大约为 6.2。
For cube roots, the method is similar. Estimate ∛50 by recognising that 3³ = 27 and 4³ = 64, so ∛50 is between 3 and 4, closer to 4, perhaps 3.7. Practice improves accuracy.
估算立方根的方法类似。估算 ∛50 时，注意 3³ = 27，4³ = 64，因此 ∛50 在 3 与 4 之间，更靠近 4，大约为 3.7。多加练习能提高准确性。
You can also use trial and improvement: guess a number, cube or square it, compare with the target, and adjust your guess accordingly.
你也可以采用尝试与修正的方法：先猜一个数，求它的立方或平方，与目标比较，再相应调整猜测值。
10. Applying Powers and Roots in Problems | 幂与根在实际问题中的应用
Powers and roots appear in many real-life contexts. For example, area of a square is side length squared (A = s²), and volume of a cube is side length cubed (V = s³). If you know the area, the side length is √A.
幂与根常出现在许多生活场景中。例如，正方形的面积等于边长平方（A = s²），立方体的体积等于边长立方（V = s³）。如果已知面积，边长即为 √A。
Problems involving reverse percentages, speed, density or pressure may also require you to find roots when a quantity is squared or cubed in a formula. Understanding how to rearrange and solve such equations is an essential part of Key Stage 3.
涉及逆百分比、速度、密度或压力的问题，若公式中含有平方或立方项，也可能需要求平方根或立方根。理解如何变形并求解此类方程是 KS3 的重要内容。
In addition, powers are used when writing very large or very small numbers in standard form (also called scientific notation), which you will encounter later. Mastery of powers and roots now will prepare you for future topics.
此外，当以标准形式（也称科学记数法）书写特别大或特别小的数时也会用到幂，你将在以后的学习中接触。现在熟练掌握幂与根将为未来的学习打好基础。
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