Indices, surds and standard form · 指数、根式与标准形式
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| index/ˈɪndeks/ | 指数 | zhǐ shù |
How small is a microscopic length?
- A microscope records a length of 0.000072 metres. A compact representation must keep its size correct.
- This lesson studies index 指数: The power to which a base is raised.
微观长度有多小？
- 显微镜测得长度为 0.000072 米。紧凑表示法必须保持其数值准确。
- 本课研究 指数 指数: The power to which a base is raised.
Choose the mathematical structure
- For the same positive base, multiplication adds indices and division subtracts them. A negative index means reciprocal; a fractional index represents a root. Standard form has 1≤a<10.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
选择数学结构
- 对于同一个正底数，乘法对应指数相加，除法对应指数相减。负指数意味着倒数；分数指数代表开方。科学计数法满足 1≤a<10。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
Which description correctly defines index? · 下列哪项描述正确定义了指数？
The power to which a base is raised. · 底数被提升的幂次。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
0.000072=7.2×10^(-5). Also 16^(3/4)=(16^(1/4))^3=2^3=8. Simplify √72=6√2, then rationalise 1/√2=√2/2.
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
0.000072=7.2×10^(-5)。此外 16^(3/4)=(16^(1/4))^3=2^3=8。化简 √72=6√2，然后对 1/√2=√2/2 进行分母有理化。
Indices, surds and standard form · 指数、根式与标准形式
For the same positive base, multiplication adds indices and division subtracts them · 相同正底数相乘时指数相加，相除时指数相减
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Evaluate 16^(3/4). · 计算 16^(3/4)。
16=2⁴, so 16^(3/4)=2³=8. · 因16=2⁴，故16^(3/4)=2³=8。
Test a tempting shortcut
- Index laws do not turn a sum into a single power: 2^3+2^4=24, not 2^7. Do not round a surd when an exact answer is requested.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
For every positive a, a^2+a^3 equals a^5. This claim is false. Explain which definition or assumption it violates.
检验一个诱人的捷径
- 指数法则不能将和转化为单一幂：2^3+2^4=24，而非2^7。若要求精确答案，切勿对根式进行四舍五入。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
对于任意正数a，a^2+a^3等于a^5。此说法错误。请说明它违反了哪一定义或假设。
Evaluate 2^(-3). · 计算 2^(-3)。
A negative index gives a reciprocal: 2^(-3)=1/8=0.125. · 负指数表示倒数：2^(-3)=1/8=0.125。
For every positive a, a^2+a^3 equals a^5. · 对于任意正数a，a^2+a^3等于a^5。
Index laws do not turn a sum into a single power: 2^3+2^4=24, not 2^7. Do not round a surd when an exact answer is requested. · 指数法则不能将求和转化为单一幂次：2^3+2^4=24，而非2^7。若要求精确答案，请勿对根式进行四舍五入。
Interpret a new situation
- Check powers of ten against the original quantity. Use surds for exact geometry, and round only the final length when the question asks for a decimal.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
解读新情境
- 用十的幂次核对原始数量。在几何中应使用根式以求得精确值，仅当题目要求小数时才对最终长度进行四舍五入。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
Write 7.2×10^(-5) as a decimal. · 将 7.2×10^(-5) 写成小数。
Move the decimal point five places left: 7.2×10^(-5)=0.000072. · 小数点左移五位：7.2×10^(-5)=0.000072。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- 7357 · A-level · B. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The power to which a base is raised. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- 7357 · A-Level · B. 在分配扩展内容前，请匹配目标层级与教学大纲。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
底数被提升的次数（即指数）。请选择正确的关系，展示推导方法，检查其适用条件并解释结果。