If $\omega$ ($\neq 1$) is a cube root of unity and $(1 + \omega)^7 = A + B\omega$, then $A$ and $B$ are respectively
Step-by-Step Solution
Key Concept: $(1+\omega)^7=(-\omega^2)^7=-\omega^{14}=-\omega^2=1+\omega$, giving $A=B=1$. The key substitution is $1+\omega=-\omega^2$.
**Step 1: Simplify 1+ω**
$1+\omega+\omega^2=0 \Rightarrow 1+\omega = -\omega^2$.
**Step 2: Compute the 7th power**
$(-\omega^2)^7 = -\omega^{14} = -\omega^{14\bmod3} = -\omega^2$.
**Step 3: Express in terms of ω**
$-\omega^2 = -(- 1-\omega) = 1+\omega$. So $A+B\omega = 1+\omega \Rightarrow A=1,\,B=1$.
Correct Answer: 2