If ω (≠1) is a cube root of unity and (1+ω)7=A+Bω then A and B are respectively.
0,1
1,1
1,0
−1,1
Apply the properties of the cube root of unity of complex numbers.
∴ω3=1and1+ω+ω2=0
Now, (1+ω)7=A+Bω
⇒(−ω2)7=A+Bω∵1+ω=−ω2⇒(−ω14)=A+Bω∵(an)m=an×m⇒−(ω3×ω3×ω3×ω3×ω2)=A+Bω[∵ω3=1]⇒−(ω2)=A+Bω⇒1+ω=A+Bω
Now, compare the left-hand side to the right-hand side, and we get
A=1andB=1
Hence, the correct answer is an option (B).
If ω≠1 is a cube root of unity and (1+ω)7=A+Bω then AandB are respectively