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If-z-1-3-3-3-i-and-z-2-1-3-i-Find-z-1-3-z-2-6-




Question Number 203309 by hardmath last updated on 15/Jan/24
If   z_1 = 3 + 3(√3) i   and   z_2 = -1 − (√3) i  Find:   (z_1 ^( 3) /z_2 ^( 6) ) = ?
Ifz1=3+33\boldsymboliandz2=13\boldsymboliFind:z13z26=?
Answered by Rasheed.Sindhi last updated on 16/Jan/24
z_1 =3(1+(√3) i) ∧ z_2 =−(1+(√3) i)  ⇒z_1 =−3z_2   (z_1 ^( 3) /z_2 ^( 6) ) =(((−3z_2 )^3 )/z_2 ^6 )=((−27z_2 ^3 )/z_2 ^6 )=((−27)/z_2 ^3 )     z_2 +1=−(√3) i⇒z_2 ^2 +2z_2 +1=−3  z_2 ^2 =−2z_2 −4⇒z_2 ^3 =−2z_2 ^2 −4z_2    =−2(−2z_2 −4)−4z_2 =8  (z_1 ^( 3) /z_2 ^( 6) ) =((−27)/z_2 ^3 )=((−27)/8)=−((27)/8)
z1=3(1+3i)z2=(1+3i)z1=3z2z13z26=(3z2)3z26=27z23z26=27z23z2+1=3iz22+2z2+1=3z22=2z24z23=2z224z2=2(2z24)4z2=8z13z26=27z23=278=278
Commented by hardmath last updated on 19/Jan/24
thank you dear ser
thankyoudearser
Answered by behi834171 last updated on 15/Jan/24
z_1 =3(1+(√3)i),z_2 =−(1+(√3)i),z_1 =−3z_2   z_2 =−2((1/2)+((√3)/2)i)=−2(cos(𝛑/3)+i.sin(𝛑/3))=  =−2.e^(i.(𝛑/3)) ⇒z_2 ^3 =[−2e^(i.(𝛑/3)) ]^3 =−8e^(i.𝛑) =+8  ⇒(z_1 ^3 /z_2 ^6 )=((z_1 /z_2 ))^3 .(1/z_2 ^3 )=(−3)^3 .(1/8)=−((27)/( 8))  .■  [note!     e^(i.𝛑) =cos𝛑+i.sin𝛑=−1]
z1=3(1+3i),z2=(1+3i),z1=3z2\boldsymbolz2=2(12+32\boldsymboli)=2(\boldsymbolcos\boldsymbolπ3+\boldsymboli.\boldsymbolsin\boldsymbolπ3)==2.\boldsymbole\boldsymboli.\boldsymbolπ3\boldsymbolz23=[2\boldsymbole\boldsymboli.\boldsymbolπ3]3=8\boldsymbole\boldsymboli.\boldsymbolπ=+8\boldsymbolz13\boldsymbolz26=(\boldsymbolz1\boldsymbolz2)3.1\boldsymbolz23=(3)3.18=278.◼[\boldsymbolnote!\boldsymbole\boldsymboli.\boldsymbolπ=\boldsymbolcosπ+\boldsymboli.\boldsymbolsinπ=1]
Commented by hardmath last updated on 19/Jan/24
thank you dear ser
thankyoudearser
Answered by Rasheed.Sindhi last updated on 16/Jan/24
z_1 −3=3(√3) i  z_1 ^2 −6z_1 +9=−27⇒z_1 ^2 =6z_1 −36  ⇒z_1 ^3 =6z_1 ^2 −36z_1 =6(6z_1 −36)−36z_1 =−216     z_2 +1=−(√3) i  z_2 ^2 +2z_2 +1=−3⇒z_2 ^2 =−2z_2 −4  ⇒z_2 ^3 =−2z_2 ^2 −4z_2 =−2(−2z_2 −4)−4z_2 =8   ⇒z_2 ^6 =64     ▶(z_1 ^3 /z_2 ^6 )=((−216)/(64))=−((27)/8)
z13=33iz126z1+9=27z12=6z136z13=6z1236z1=6(6z136)36z1=216z2+1=3iz22+2z2+1=3z22=2z24z23=2z224z2=2(2z24)4z2=8z26=64z13z26=21664=278

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