Rational indices and real domains
| English | 中文 | Pinyin |
|---|---|---|
| rational exponent/ˈræʃənl ekˈspəʊnənt/ | 有理指数 | yǒu lǐ zhǐ shù |
A cube grows to 27 times its original volume. Why does its edge grow by only a factor of 3?
- A cube grows to 27 times its original volume. Why does its edge grow by only a factor of 3?
- This lesson studies rational exponent 有理指数: An exponent written as a fraction of integers.
Choose the mathematical structure
- For a positive base a, a^(p/q) is the qth root of a raised to p, with q positive. A negative exponent takes the reciprocal, so the base must be nonzero. Reduce p/q first. For a negative real base, a reduced odd denominator permits a real root; an even denominator does not.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines rational exponent?
An exponent written as a fraction of integers.
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.
27^(2/3)=(cube root of 27)²=9, while 16^(−3/4)=1/(fourth root of 16)³=1/8. For x^(2/3)=4 over the real numbers, let u be the cube root of x: u²=4, so u=±2 and x=±8. Both inputs check. By contrast x^(1/2)=−2 has no real solution because the principal square root is nonnegative. For negative bases, (−8)^(1/3)=−2 and (−8)^(2/3)=4.
Rational indices and real domains
For a positive base a, a^(p/q) is the qth root of a raised to p, with q positive
Check the conditions behind each index or surd manipulation.
Evaluate 16^(−3/4).
The fourth root of 16 is 2; its cube is 8, then take the reciprocal.
Test a tempting shortcut
- Do not change a negative exponent into a negative answer. Do not discard a negative solution when an odd root is squared. General power-of-a-power manipulations can fail outside positive bases: ((−1)²)^(1/2)=1, whereas (−1)^(2×1/2)=−1.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The equation x^(2/3)=4 has only the solution x=8. This claim is false. Explain which definition or assumption it violates.
Find the negative real solution of x^(2/3)=4.
Let u be the cube root of x. The negative root u=−2 gives x=−8.
The equation x^(2/3)=4 has only the solution x=8.
Do not change a negative exponent into a negative answer. Do not discard a negative solution when an odd root is squared. General power-of-a-power manipulations can fail outside positive bases: ((−1)²)^(1/2)=1, whereas (−1)^(2×1/2)=−1.
Interpret a new situation
- Use a positive scale factor for lengths and volumes: a volume ratio 27 gives edge ratio 27^(1/3)=3 and area ratio 27^(2/3)=9. In equations, state the real domain and substitute every candidate. Zero to a negative power is undefined; this lesson does not assign a value to 0^0.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Evaluate (−8)^(2/3).
The real cube root of −8 is −2; its square is 4.
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
- 7357 · A-level · B. 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.
An exponent written as a fraction of integers. Choose the relationship, show the method, check its assumptions and interpret the result.