Indices, surds and standard form · Higher · Índices, radicais e notação científica · Superior
| English | Português |
|---|---|
| index/ˈɪndeks/ | interface |
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? · Qual descrição define corretamente índice?
The power to which a base is raised. · O expoente ao qual uma base é elevada.
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 · Índices, radicais e notação científica
For the same positive base, multiplication adds indices and division subtracts them · Para a mesma base positiva, a multiplicação soma os índices e a divisão os subtrai.
Compare the model with the worked case and explain one change. · Compare o modelo com o caso resolvido e explique uma mudança.
Evaluate 16^(3/4). · Calcule 16^(3/4).
16=2⁴, so 16^(3/4)=2³=8. · 16=2⁴, logo 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). · Avalie 2^(-3).
A negative index gives a reciprocal: 2^(-3)=1/8=0.125. · Um índice negativo resulta no inverso: 2^(-3)=1/8=0.125.
For every positive a, a^2+a^3 equals a^5. · Para qualquer positivo a, a^2+a^3 é igual a 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. · As leis dos índices não transformam uma soma em uma única potência: 2^3+2^4=24, não 2^7. Não arredonde um radical quando for solicitado uma resposta exata.
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. · Escreva 7.2×10^(-5) como um decimal.
Move the decimal point five places left: 7.2×10^(-5)=0.000072. · Desloque o ponto decimal cinco posições para a esquerda: 7.2×10^(-5)=0.000072.
Match each part of a complete solution to its purpose. · Associe cada parte de uma solução completa ao seu propósito.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Uma suposição justifica o modelo; uma verificação testa o resultado; a interpretação conecta-o à pergunta.
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.