Answer to Question #107109 in Algebra for Chinmoy Kumar Bera

Question #107109
obtain the 6th root of -7 and represent them in argand diagram
1
Expert's answer
2020-04-01T14:08:15-0400


76\sqrt[6]{-7} we need to write the number -7 in trigonometric form: 7=7(cosπ+іsinπ)-7=7(cos π + і sin π) zk=76(cos(π/6+(2π/6)k)+isin(π/6+(2π/6)k)),k=0,1,2,3,4,5z_k=\sqrt[6]{7}(cos⁡(π/6+(2π/6) k)+i sin⁡(π/6+(2π/6) k)), k=0,1,2,3,4,5

from here

z0=76(cos(π/6)+isin(π/6))=(763)/2+(76/2)iz_0=\sqrt[6]{7}(cos⁡(π/6)+i sin⁡(π/6))=(\sqrt[6]{7}\sdot\sqrt{3})/2+(\sqrt[6]{7}/2)i\\ ;

z1=76(cos(π/2)+isin(π/2))=76iz_1=\sqrt[6]{7}(cos⁡(π/2)+i sin⁡(π/2))=\sqrt[6]{7}i\\ ;

z2=76(cos(5π/6)+isin(5π/6))=(763)/2+(76/2)iz_2=\sqrt[6]{7}(cos⁡(5π/6)+i sin⁡(5π/6))=-(\sqrt[6]{7}\sdot\sqrt{3})/2+(\sqrt[6]{7}/2)i\\ ;

z3=76(cos(5π/6)+isin(5π/6))=(763)/2(76/2)iz_3=\sqrt[6]{7}(cos⁡(5π/6)+i sin⁡(5π/6))=-(\sqrt[6]{7}\sdot\sqrt{3})/2-(\sqrt[6]{7}/2)i\\;

z4=76(cos(3π/2)+isin(3π/2))=76iz_4=\sqrt[6]{7}(cos⁡(3π/2)+i sin⁡(3π/2))=-\sqrt[6]{7}i\\;

z5=76(cos(5π/6)+isin(5π/6))=(763)/2(76/2)iz_5=\sqrt[6]{7}(cos⁡(5π/6)+i sin⁡(5π/6))=(\sqrt[6]{7}\sdot\sqrt{3})/2-(\sqrt[6]{7}/2)i\\ ;



Answer: z0=(763)/2+(76/2)i;z_0=(\sqrt[6]{7}\sdot\sqrt{3})/2+(\sqrt[6]{7}/2)i;

z1=76iz_1=\sqrt[6]{7}i ;

z2=(763)/2+(76/2)iz_2=-(\sqrt[6]{7}\sdot\sqrt{3})/2+(\sqrt[6]{7}/2)i;

z3=(763)/2(76/2)iz_3=-(\sqrt[6]{7}\sdot\sqrt{3})/2-(\sqrt[6]{7}/2)i;

z4=76iz_4=-\sqrt[6]{7}i;

z5=(763)/2(76/2)iz_5=(\sqrt[6]{7}\sdot\sqrt{3})/2-(\sqrt[6]{7}/2)i.




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