실수, 유리 지수 및 복소 연산
도입됨| English | 의미 |
|---|---|
| conjugate/ˈkɒndʒuːɡeɪt/ | 복소수 켤레 |
| rational exponent/ˈræʃənl ekˈspəʊnənt/ | 유리 지수 |
A decision before an answer
- The sum of two irrational numbers can be rational: √2 + (−√2) = 0. A rule about individual terms need not survive their combination.
- Your goal: Distinguish rational and irrational numbers without assuming their sums retain type.
Read the relationship
- A rational number is a ratio of integers with nonzero denominator; a terminating or repeating decimal is rational. An irrational real number is not such a ratio. Closure of rational numbers under addition and multiplication does not imply closure of irrational numbers. Exact roots may simplify: √18=3√2 remains irrational, whereas √16=4 is rational.
- Apply exponent and radical rules on the permitted domain.
What is (3+i)(3-i)?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
Conjugate product is 3²+1²=10.
Use the defining rule
- For a positive base, a^(m/n) combines an nth root and an integer power. A negative exponent forms a reciprocal and requires a nonzero base. Distinguish √(x²)=|x| from x. When an even root is involved, respect the real domain and the principal nonnegative root. Distribute powers across products, not across sums: (a+b)² includes the cross term.
- Calculate with complex numbers using i²=-1 and conjugates.
81^(1/2) equals:
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
The principal square root is 9.
Check the conditions
- Complex arithmetic uses i²=-1. Add real parts and imaginary parts separately, and multiply by distribution before reducing i². To divide by a+bi, multiply numerator and denominator by a-bi; the resulting denominator is a²+b² when the original is nonzero. Complex roots permit solutions to equations such as x²=-9 that have no real roots.
- Calculate with complex numbers using i²=-1 and conjugates.
16^(3/4)=2³=8. (2+3i)(1-2i)=2-4i+3i-6i²=8-i. For (1+i)/(1-i), multiplying by 1+i gives (1+i)²/2=2i/2=i. The solutions of x²+9=0 are ±3i; neither is real.
The real part of (4+2i)+(1-5i) is ____.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
Real parts add to 4+1; imaginary part is -3i.
Apply the task format
- Enhanced ACT still includes selected advanced topics, and no formula sheet is supplied. Build familiarity with the defining rules rather than guessing from a calculator display. Check whether the question asks for real or complex solutions, an exact radical or a decimal approximation. The local checks here are formative, not an official scored form.
- Calculate with complex numbers using i²=-1 and conjugates.
Do not infer that two irrational terms must have an irrational sum, distribute a square over addition, or replace √(x²) by x when x can be negative.
다음 중 이 사례에 부합하는 답안은 무엇입니까?
有理수和无理数를 구분하되, 두 수의 합이 동일한 유형을 유지한다고 가정하지 마십시오.
The sum of two irrational numbers is always irrational.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
A number plus its negative gives rational zero.
Keep the distinctions
- conjugate 共轭复数 — The number a-bi paired with a+bi.
- rational exponent 有理数指数 — An exponent m/n interpreted using powers and roots on the stated domain.
- Distinguish rational and irrational numbers without assuming their sums retain type.
- Apply exponent and radical rules on the permitted domain.
- Calculate with complex numbers using i²=-1 and conjugates.
Match each term with its precise meaning in this lesson.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
난이도 3/5 · 200 코인 + 오늘 첫 완료 시 소액 보너스
The order follows the stated process; check each stage before the next.