1.7-bo'lim : Murakkab raqamlar
9. Ko'rsatilgan amalni bajaring va javobingizni standart shaklda yozing.
\[\frac{{6 + 7i}}{{8 - i}}\]
Maslahat: Esda tutingki, standart shakl maxrajda \(i\) ga ruxsat bermaydi.
Yechim
Standart shakl \(i\) ning maxrajda bo'lishiga yo'l qo'ymagani uchun biz pay va maxrajni maxrajning konjugatiga, ya'ni \(8 + i\)ga ko'paytirishimiz kerak bo'ladi.
2-qadam
Konjugatga ko'paytirish,
\[\frac{{6 + 7i}}{{8 - i}}\,\,\frac{{8 + i}}{{8 + i}} = \frac{{\left( {6 + 7i} \right)\left( {8 + i} \right)}}{{\left( {8 - i} \right)\left( {8 + i} \right)}}\]3-qadam
Endi biz qilishimiz kerak bo'lgan yagona narsa surat va maxrajdagi ko'paytirishni bajarish va natijani standart shaklga qo'yishdir.
\[\frac{{6 + 7i}}{{8 - i}}\,\,\frac{{8 + i}}{{8 + i}} = \frac{{48 + 62i + 7{i^2}}}{{64 - {i^2}}} = \frac{{41 + 62i}}{{65}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{41}}{{65}} + \frac{{62}}{{65}}i}}\]