Integer indices and standard form · Foundation
| English | 中文 | Pinyin |
|---|---|---|
| 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?
The power to which a base is raised.
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
Use integer powers, square and cube roots and standard form with 1≤a<10
Compare the model with the worked case and explain one change.
Evaluate 2³×2⁴.
Same base: add indices. 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.
2³=2×2×2=8.
For every positive a, a^2+a^3 equals 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.
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.
Move the decimal point five places left: 7.2×10^(-5)=0.000072.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
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.