Question

If $\omega$ is complex cube root of unity and $(1+\omega)^7=A+B\omega$, then values of $A$ and $B$ are, respectively

• A. 0, 1
• B. 1, 1
• C. 1, 0
• D. -1, 1

Hint:

Whenever you see $(1+\omega)$ or $(1+\omega^2)$, immediately replace them with $-\omega^2$ and $-\omega$, respectively. This single substitution rapidly collapses high-power polynomials into easily manageable terms.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to simplify the expression $(1+\omega)^7$ using the properties of the complex cube roots of unity to find the integer coefficients $A$ and $B$.

Step 2: Key Formula or Approach:
The complex cube roots of unity ($1, \omega, \omega^2$) satisfy two fundamental properties:
7. $1 + \omega + \omega^2 = 0$
8. $\omega^3 = 1$

Step 3: Detailed Explanation:
From the first property, we can rearrange the terms to isolate $(1 + \omega)$:
$$1 + \omega = -\omega^2$$
Now substitute this back into the original expression:
$$(1+\omega)^7 = (-\omega^2)^7$$
$$= (-1)^7 \cdot (\omega^2)^7$$
$$= -1 \cdot \omega^{14}$$
Now, simplify $\omega^{14}$ using the property $\omega^3 = 1$. Divide 14 by 3 to find the remainder: $14 = 3 \times 4 + 2$.
$$\omega^{14} = (\omega^3)^4 \cdot \omega^2 = (1)^4 \cdot \omega^2 = \omega^2$$
So, the expression reduces to $-\omega^2$.
We need it in the form $A + B\omega$. Again, use the property $1 + \omega + \omega^2 = 0 \implies -\omega^2 = 1 + \omega$.
Therefore, $(1+\omega)^7 = 1 + \omega$.
Comparing this to $A + B\omega$, we see that $A = 1$ and $B = 1$.

Step 4: Final Answer:
The values are $1, 1$, matching option (B).