Integer indices and standard form · Foundation · 整数指数与标准形式 · 基础
| 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
- Use integer powers, square and cube roots and standard form with 1≤a<10. In multiplying powers with the same base, add indices; in division, subtract them.
- 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
- 使用整数指数、平方根和立方根以及科学计数法（其中 1≤a<10）。同底数幂相乘时指数相加；相除时指数相减。
- 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 2³×2⁴=2⁷=128. The square root of 81 is 9. Check a standard-form answer by writing it out as a decimal.
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 2³×2⁴=2⁷=128。81的平方根是9。通过将科学计数法答案写成小数形式来验证其正确性。
Integer indices and standard form · 整数指数与标准形式
Use integer powers, square and cube roots and standard form with 1≤a<10 · 使用整数幂、平方根和立方根以及标准形式，其中 1≤a<10
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 2³×2⁴. · 计算 2³×2⁴。
Same base: add indices. 2³×2⁴=2⁷=128. · 底数相同：指数相加。2³×2⁴=2⁷=128。
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 2³. · 计算 2³。
2³=2×2×2=8.
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
- This Foundation/Core lesson excludes fractional powers and surd rationalisation. Estimate a result before using a calculator and retain the required precision.
- 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
- 本基础/核心课程不包含分数指数和根式分母有理化。在使用计算器之前先估算结果，并保留所需的精度。
- 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
- 4MA1 · Foundation · 1. 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
- 4MA1 · 基础级 · 1. 在分配拓展内容前，请匹配目标层级和教学大纲要求。
- 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。