2
Ildizlar va ratsional ko‘rsatkichli daraja
maktab darsligi: algebra 8-sinf, kvadrat ildiz; algebra 9-sinf va matematika 10-sinf, n-darajali ildiz
Maqsad: Ildiz xossalarini ishlatish va ildizni darajaga aylantirish.
Qoida va g‘oya
Arifmetik kvadrat ildiz sqrt(a) (a ≥ 0) — kvadrati a ga teng manfiy bo‘lmagan son. Muhim: sqrt(a^2) = |a|. Xossalar: sqrt(ab) = sqrt(a) · sqrt(b); sqrt(a/b) = sqrt(a) / sqrt(b); (sqrt(a))^2 = a. n-darajali ildiz uchun ham shunday; toq n da manfiy sondan ildiz olinadi: kub ildiz (−27) = −3, juft n da faqat a ≥ 0. Ratsional ko‘rsatkich: a^(m/n) = n-darajali ildiz (a^m), masalan 8^(2/3) = (kub ildiz 8)^2 = 4, a^(1/2) = sqrt(a). Ildiz ostidan ko‘paytuvchini chiqarish: sqrt(72) = sqrt(36 · 2) = 6 sqrt(2).
Namunaviy masalalar
sqrt(72) ni soddalashtiring.
Yechim: 72 = 36 · 2, sqrt(72) = 6 sqrt(2).
Javob: 6 sqrt(2)
Yechim: 72 = 36 · 2, sqrt(72) = 6 sqrt(2).
Javob: 6 sqrt(2)
27^(2/3) ni hisoblang.
Yechim: Kub ildiz 27 = 3, 3^2 = 9.
Javob: 9
Yechim: Kub ildiz 27 = 3, 3^2 = 9.
Javob: 9
sqrt((−5)^2) + kub ildiz(−8) ni hisoblang.
Yechim: sqrt(25) = 5, kub ildiz(−8) = −2. 5 + (−2) = 3.
Javob: 3
Yechim: sqrt(25) = 5, kub ildiz(−8) = −2. 5 + (−2) = 3.
Javob: 3
Tipik xatolar
- sqrt(a^2) = a deb yozish: to‘g‘risi |a|, shuning uchun sqrt((−5)^2) = 5.
- sqrt(a + b) = sqrt(a) + sqrt(b) deb olish: bunday xossa yo‘q.
- Juft darajali ildizni manfiy sondan olish: sqrt(−4) haqiqiy son emas.
Mashq
1
sqrt(50) · sqrt(2) ni hisoblang.
10
2
32^(1/5) ni hisoblang.
2
3
81^(3/4) ni hisoblang.
27
4
sqrt(98) ni a·sqrt(b) ko‘rinishida yozing.
7 sqrt(2)