1.3-bo'lim : Radikallar
12. Quyidagi ifodani soddalashtiring. \(x\), \(y\) va \(z\) ijobiy deb faraz qiling.
\[\sqrt[4]{{4{x^3}y}}\,\,\sqrt[4]{{8{x^2}{y^3}{z^5}}}\]Yechim
Esda tutingki, ifodada bir xil indeksli ikkita radikalning mahsuloti bo'lsa, soddalashtirish jarayonini boshlashdan oldin ularni birinchi navbatda bitta ildizga birlashtirishimiz kerak.
\[\sqrt[4]{{4{x^3}y}}\,\,\sqrt[4]{{8{x^2}{y^3}{z^5}}} = \,\sqrt[4]{{\left( {4{x^3}y} \right)\left( {8{x^2}{y^3}{z^5}} \right)}} = \,\sqrt[4]{{32{x^5}{y^4}{z^5}}}\]2-qadam
Endi ifoda bitta radikal sifatida yozilgan bo'lsa, biz oldingi masalalarda bo'lgani kabi davom etishimiz mumkin.
Radikand quyidagicha yozilishi mumkin:
\[32{x^5}{y^4}{z^5} = \left( {{2^4}{x^4}{y^4}{z^4}} \right)\left( {2xz} \right)\]3-qadam
Endi biz radikalni qayta yozganimizdan so'ng, radikal bilan kurashish va ifodani soddalashtirilgan radikal shaklda olish oson.
\[\sqrt[4]{{4{x^3}y}}\,\,\sqrt[4]{{8{x^2}{y^3}{z^5}}} = \,\sqrt[4]{{32{x^5}{y^4}{z^5}}} = \sqrt[4]{{{2^4}{x^4}{y^4}{z^4}}}\,\,\sqrt[4]{{2xz}} = \require{bbox} \bbox[2pt,border:1px solid black]{{2xyz\,\,\sqrt[4]{{2xz}}}}\]