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Algebra II

Course: Algebra II > Unit 2

Lesson 1: Mavhum sonlar nima?

Mavhum birlikning darajalari

Mavhum birlik i ning darajasini qanday soddalashtirishni oʻrganing. Masalan, i²⁷ ni -i kabi soddalashtiring.

Bilamizki, i, equals, square root of, minus, 1, end square root va i, squared, equals, minus, 1.

Biroq i, cubed va i, start superscript, 4, end superscript haqida-chi? i ning boshqa butun darajalari-chi? Bularni qanday hisoblaymiz?

i, cubed va i, start superscript, 4, end superscriptni topish

Darajalar xossasi bu yerda yordam berishi mumkin! i ning darajasini hisoblayotganda biz toʻgʻri deb bilgan darajalarning xossalarini haqiqiy sonlar tizimida qoʻllashimiz mumkin, agar darajalar butun son boʻlsa.

Shunga asoslanib, i, cubed va i, start superscript, 4, end superscript ni topamiz.

Bizga maʼlumki, i, cubed, equals, i, squared, dot, i. Ammo i, squared, equals, minus, 1 boʻlar ekan, biz shuni kuzatishimiz mumkin:

$\begin{aligned} i^3 &= {{i^2}}\cdot i\\ \\ &={ (-1)}\cdot i\\ \\ &= \purpleD{-i} \end{aligned}$

Shuningdek, i, start superscript, 4, end superscript, equals, i, squared, dot, i, squared. Yana i, squared, equals, minus, 1 dan foydalanib, quyidagi natijaga erishamiz:

$\begin{aligned} i^4 &= {{i^2\cdot i^2}}\\ \\ &=({ -1})\cdot ({-1})\\ \\ &= \goldD{1} \end{aligned}$

i ning darajalari

Shu yoʻsinda davom etamiz! Bizga tanish usuldan foydalanib i ning navbatdagi 4 ta darajasini topamiz:

$\begin{aligned} \Large i^5 &= {i^4\cdot i}~~~~~&&\small{\gray{\text{Daraja xossalari}}}\\ \\ &=1\cdot i&&\small{\gray{\text{ $i^4=1$ boʻlgani uchun}}}\\ \\ &= \blueD i \end{aligned}$

$\begin{aligned}\Large i^6 &= {i^4\cdot i^2}&&\small{\gray{\text{Daraja xossalari}}}\\ \\ &=1\cdot (-1)&&\small{\gray{\text{$i^4=1$ va $i^2=-1$ boʻlgani uchun}}}\\ \\ &=\greenD{-1} \end{aligned}$

$\begin{aligned}\Large i^7 &= {i^4\cdot i^3}&&\small{\gray{\text{Daraja xossalari}}}\\ \\ &=1\cdot (-i)&&\small{\gray{\text{ $i^4=1$ va $i^3=-i$ boʻlgani uchun}}}\\ \\ &=\purpleD{-i} \end{aligned}$

$\begin{aligned}\Large i^8 &= {i^4\cdot i^4~~~~}&&\small{\gray{\text{Daraja xossalari}}}\\ \\ &=1\cdot 1&&\small{\gray{\text{$i^4=1$ boʻlgani uchun}}}\\ \\ &=\goldD 1 \end{aligned}$

Natijalarni jadvalda yozamiz:

i, start superscript, 1, end superscript	i, squared	i, cubed	i, start superscript, 4, end superscript	i, start superscript, 5, end superscript	i, start superscript, 6, end superscript	i, start superscript, 7, end superscript	i, start superscript, 8, end superscript
start color #11accd, i, end color #11accd	start color #1fab54, minus, 1, end color #1fab54	start color #7854ab, minus, i, end color #7854ab	start color #e07d10, 1, end color #e07d10	start color #11accd, i, end color #11accd	start color #1fab54, minus, 1, end color #1fab54	start color #7854ab, minus, i, end color #7854ab	start color #e07d10, 1, end color #e07d10

Darajalarning ketma-ketligi

Jadvaldan, i ning darajalari start color #11accd, i, end color #11accd, start color #1fab54, minus, 1, end color #1fab54, start color #7854ab, minus, i, end color #7854ab va start color #e07d10, 1, end color #e07d10 ketma-ketlikda ketishi koʻrinadi.

Bu ketma-ketlikdan foydalanib i, start superscript, 20, end superscript topa olamizmi? Keling, urinib koʻramiz.

Quyidagi qatorda dastlabki 20 ta son keltirilgan:

\quad

start color #11accd, i, end color #11accd, start color #1fab54, minus, 1, end color #1fab54, start color #7854ab, minus, i, end color #7854ab, start color #e07d10, 1, end color #e07d10, start color #11accd, i, end color #11accd, start color #1fab54, minus, 1, end color #1fab54, start color #7854ab, minus, i, end color #7854ab, start color #e07d10, 1, end color #e07d10, start color #11accd, i, end color #11accd, start color #1fab54, minus, 1, end color #1fab54, start color #7854ab, minus, i, end color #7854ab, start color #e07d10, 1, end color #e07d10, start color #11accd, i, end color #11accd, start color #1fab54, minus, 1, end color #1fab54, start color #7854ab, minus, i, end color #7854ab, start color #e07d10, 1, end color #e07d10, start color #11accd, i, end color #11accd, start color #1fab54, minus, 1, end color #1fab54, start color #7854ab, minus, i, end color #7854ab, start color #e07d10, 1, end color #e07d10

Bu mantiqqa koʻra, i, start superscript, 20, end superscript start color #e07d10, 1, end color #e07d10ga teng boʻlishi kerak. Keling, darajalar yordamida buni koʻrib chiqamiz. Yodingizda boʻlsin, biz bu yerda darajalarning xossalaridan, xuddi haqiqiy sonlar bilan boʻlgani kabi foydalanishimiz mumkin!

$\begin{aligned} i^{20} &= (i^4)^5&&\small{\gray{\text{Daraja xossalari}}}\\ \\ &= (1)^5 &&\small{\gray{i^4=1}}\\\\ &= \goldD 1 &&\small{\gray{\text{Soddalashtiring}}}\end{aligned}$

Har qanday holatda ham i, start superscript, 20, end superscript, equals, 1 ekanini koʻramiz.

i ning darajalari

Biz endi i, start superscript, 138, end superscriptni topishga urinib koʻraylik. Biz ketma-ketlik roʻyxatini 138, start superscript, start text, c, h, i, end text, end superscript hadda tuzishimiz mumkin start color #11accd, i, end color #11accd, start color #1fab54, minus, 1, end color #1fab54, start color #7854ab, minus, i, end color #7854ab, start color #e07d10, 1, end color #e07d10,... , ammo bu juda koʻp vaqt oladi!

Ahamiyat bering, i, start superscript, 4, end superscript, equals, 1, i, start superscript, 8, end superscript, equals, 1, i, start superscript, 12, end superscript, equals, 1 va hokazo. Boshqacha aytganda, i ning darajasi 4 ning karralisi boʻlganda 1ga teng boʻladi.

Biz ushbu maʼlumotdan i, start superscript, 138, end superscript ni soddalashtirishga yordam berish maqsadida uni darajalarning xossalari bilan birga qoʻllashimiz mumkin.

Misol

i, start superscript, 138, end superscript ni soddalashtiring.

Yechim

138 soni 4 ning karralisi emas, 136 esa karrali! Bundan foydalanib i, start superscript, 138, end superscriptni soddalashtiramiz.

$\begin{aligned} i^{138} &=i^{136}\cdot i^2 &&\small{\gray{\text{Daraja xossalari}}}\\\\ &=(i^{4\cdot 34})\cdot i^2&&\small{\gray{136=4\cdot 34}} \\\\ &=(i^{4})^{34}\cdot i^2&&\small{\gray{\text{Daraja xossalari}}} \\\\ &=(1)^{34}\cdot i^2 &&\small{\gray{\text{$i^4=1$}}}\\\\ &=1\cdot -1&&\small{\gray{\text{$i^2=-1$}}}\\\\ &=-1 \end{aligned}$

Shu bois i, start superscript, 138, end superscript, equals, minus, 1.

Endi nega i, start superscript, 138, end superscript ifodani i, start superscript, 136, end superscript, dot, i, squared koʻrinishga keltirganimizni soʻrashingiz mumkin.

Agar asl daraja 4ga karrali boʻlmasa, u holda undan kichikroq boʻlgan eng yaqin karralisini topish darajani, shunchaki i, start superscript, 4, end superscript, equals, 1 haqidagi maʼlumotdan foydalanib, i, i, squared, yoki i, cubed gacha soddalashtirish imkonini beradi.