Rational indices and real domains
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| rational exponent/ˈræʃənl ekˈspəʊnənt/ | số mũ hữu tỷ |
Why does its edge grow by only a factor of 3?
- A cube grows to 27 times its original volume.
- This lesson studies rational exponent 有理指数: An exponent written as a fraction of integers.
The method
- 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.
Why does x^(2/3)=4 have TWO real solutions while x^(1/2)=-2 has none?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
With u=∛x, u²=4 gives u=±2, hence x=±8 - both real because odd roots accept negatives. The square root is principal and nonnegative, so it can never equal -2.
A worked case
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).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The fourth root of 16 is 2; its cube is 8, then take the reciprocal.
Where it goes wrong
- 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.
Not true: The equation x^(2/3)=4 has only the solution x=8. x^(2/3)=4 means (x^(1/3))²=4, so the cube root is ±2 and x=8 or x=−8.
Find the negative real solution of x^(2/3)=4.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Let u be the cube root of x. The negative root u=−2 gives x=−8.
X^(2/3)=4 means (x^(1/3))²=4, so the cube root is ±2 and x=8 or x=−8.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. x^(2/3)=4 means (x^(1/3))²=4, so the cube root is ±2 and x=8 or x=−8.
Applying it
- 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.
Evaluate (−8)^(2/3).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The real cube root of −8 is −2; its square is 4.
Match each rational exponent to its evaluation.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The denominator of a reduced fraction takes the root first; a negative exponent reciprocates; even denominators forbid negative bases in real work.
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.
- Rational exponent: An exponent written as a fraction of integers.
- For a positive base a, a^(p/q) is the qth root of a raised to p, with q positive.
- Watch for: do not change a negative exponent into a negative answer.