1.3-bo'lim : Radikallar
11. Quyidagi ifodani soddalashtiring. \(x\), \(y\) va \(z\) ijobiy deb faraz qiling.
\[\sqrt[3]{{54{x^6}{y^7}{z^2}}}\]Yechim
Eslatib o'tamiz, soddalashtirish orqali biz ifodani soddalashtirilgan radikal shaklga qo'yishni xohlaymiz (buni biz ushbu bo'lim uchun eslatmalarda aniqlaganmiz).
Buning uchun bu ifoda uchun radikalni quyidagicha yozishimiz kerak:
\[54{x^6}{y^7}{z^2} = \left( {27{x^6}{y^6}} \right)\left( {2{y^1}{z^2}} \right) = {3^3}{\left( {{x^2}} \right)^3}{\left( {{y^2}} \right)^3}\left( {2y{z^2}} \right)\]2-qadam
Endi biz radikalni qayta yozganimizdan so'ng, radikal bilan kurashish va ifodani soddalashtirilgan radikal shaklda olish oson.
\[\sqrt[3]{{54{x^6}{y^7}{z^2}}} = \sqrt[3]{{{3^3}{{\left( {{x^2}} \right)}^3}{{\left( {{y^2}} \right)}^3}\left( {2y{z^2}} \right)}} = \sqrt[3]{{{3^3}{{\left( {{x^2}} \right)}^3}{{\left( {{y^2}} \right)}^3}}}\sqrt[3]{{2y{z^2}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{3{x^2}{y^2}\,\,\sqrt[3]{{2y{z^2}}}}}\]