Indices, surds and standard form
| 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
- 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.
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 16^(3/4)=(16^(1/4))^3=2^3=8. Simplify √72=6√2, then rationalise 1/√2=√2/2.
Indices, surds and standard form
For the same positive base, multiplication adds indices and division subtracts them
Compare the model with the worked case and explain one change.
Evaluate 16^(3/4).
16=2⁴, so 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.
Evaluate 2^(-3).
A negative index gives a reciprocal: 2^(-3)=1/8=0.125.
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
- 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.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.