10
Ratsional ko‘rsatkichli daraja va uning xossalari
Darslik: 42–48-betlar
MaqsadRatsional ko‘rsatkichli darajaning ta’rifi va xossalarini bilish; ularni hisoblashda qo‘llash.
Yangi so‘zlar
ratsional ko‘rsatkich · rational exponentdarajaning asosi · base of a powermanfiy ko‘rsatkich · negative exponentdarajani ildiz ko‘rinishida yozish · writing a power as a root
Tushuntirish
a > 0 va m/n ratsional son (m butun, n ≥ 2 natural) bo‘lsa, a^(m/n) = ⁿ√(aᵐ) deb ta’riflanadi; masalan, 8^(2/3) = ³√(8²) = (³√8)² = 4. Manfiy ko‘rsatkichda a^(-r) = 1/a^r. Ko‘rsatkichning kasrini qisqartirsak, daraja o‘zgarmaydi: a^(m/n) = a^(mk/(nk)). Natural ko‘rsatkichli darajaning barcha xossalari ratsional ko‘rsatkichlarda ham saqlanadi: aᵖ · aᵠ = aᵖ⁺ᵠ, aᵖ : aᵠ = aᵖ⁻ᵠ, (aᵖ)ᵠ = aᵖᵠ, (ab)ᵖ = aᵖbᵖ, (a/b)ᵖ = aᵖ/bᵖ (a > 0, b > 0). Shu xossalar yordamida ildizli ifodalarni darajalar ko‘rinishida yozib, hisoblashni soddalashtirish mumkin.
Namunalar
8^(2/3) = (³√8)² = 2² = 4; 27^(-2/3) = 1/27^(2/3) = 1/(³√27)² = 1/9.
4^(1/2) · 4^(3/2) = 4^(1/2 + 3/2) = 4² = 16; (16^(3/4)) = (⁴√16)³ = 2³ = 8.
Sinfda faoliyat
«Daraja ↔ ildiz»: jamoalar kartochkalarni juftlaydi: a^(1/2) — √a, a^(2/3) — ³√(a²), a^(-1/2) — 1/√a va hokazo.
Mashq
1
81^(3/4) ni hisoblang.
27
2
8^(4/3) ni hisoblang.
16
3
7^(2/5) · 7^(3/5) ni soddalashtiring.
7
4
Nima uchun a^(-r) = 1/a^r?
Chunki a^r · a^(-r) = a⁰ = 1, demak a^(-r) soni a^r ga teskari son.