Indices, surds and standard form · Higher
הוצג| English | משמעות |
|---|---|
| index/ˈɪndeks/ | מדריך (מעריך) |
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.
Why is 16^(3/4) equal to 8?
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A fractional index means root then power: 16^(3/4)=(⁴√16)³=2³=8. Multiplying 16 by 3/4 confuses the index with a factor.
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. 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. For a positive base, a^(m/n)=(the nth root of a)^m. Thus 27^(2/3)=3²=9 and 16^(-1/2)=1/4. To estimate √20 without a calculator, 4²<20<5² gives 4<√20<5; testing 4.5²=20.25 shows √20 is just below 4.5. Do not use a rough estimate as an exact surd answer.
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).
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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).
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A negative index gives a reciprocal: 2^(-3)=1/8=0.125.
For every positive a, a^2+a^3 equals a^5.
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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.
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Move the decimal point five places left: 7.2×10^(-5)=0.000072.
Match each index law to its correct statement.
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The laws govern products and powers of one base; they never combine a sum such as a²+a³ into a single power.
Use this in your course
- 8300 · Higher · 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.