Indices, surds and standard form
Введено| English | Значение |
|---|---|
| index/ˈɪndeks/ | показатель степени |
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.
The method
- 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.
Why is 16^(3/4) equal to 8?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
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.
A worked case
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).
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
16=2⁴, so 16^(3/4)=2³=8.
Where it goes wrong
- 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.
Not true: For every positive a, a^2+a^3 equals a^5. Index laws only add powers when the terms are multiplied: 2²+2³=12, but 2⁵=32.
Evaluate 2^(-3).
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
A negative index gives a reciprocal: 2^(-3)=1/8=0.125.
For every positive a, a^2+a^3 equals a^5.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Index laws only add powers when the terms are multiplied: 2²+2³=12, but 2⁵=32.
Applying it
- 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.
Write 7.2×10^(-5) as a decimal.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Move the decimal point five places left: 7.2×10^(-5)=0.000072.
Match each index law to its correct statement.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
The laws govern products and powers of one base; they never combine a sum such as a²+a³ into a single power.
In your course
- AQA A-level Mathematics (7357), specification section B.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Index: The power to which a base is raised.
- For the same positive base, multiplication adds indices and division subtracts them.
- Watch for: index laws do not turn a sum into a single power: 2^3+2^4=24, not 2^7.