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.
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.
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. Useful powers include 3³=27, 4³=64, 5³=125, 15²=225 and 10⁶=1,000,000. Integer laws give 2⁰=1, 2^(-3)=1/8 and (2³)²=2⁶=64. For standard-form multiplication, (3×10⁵)(4×10^(-3))=12×10²=1.2×10³. Division gives (6×10⁵)/(2×10²)=3×10³. For addition, first align exponents: 3×10⁴+2×10³=3.2×10⁴.
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.
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.
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
- 8300 · Foundation · 3.1. 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.