Key Concepts and Formulas

• Sum of Cube Roots of Unity: $1 + \omega + \omega^2 = 0$, which implies $1 + \omega = -\omega^2$.
• Cube of Unity: $\omega^3 = 1$, which implies $\omega^{3k} = 1$ for any integer $k$.
• Exponent Rules: $(ab)^n = a^n b^n$ and $(a^m)^n = a^{mn}$.

Step-by-Step Solution

Step 1: Simplify the base of the expression using the sum of roots property. We want to simplify the expression $(1+\omega)^7$. Using the property $1 + \omega + \omega^2 = 0$, we can rewrite $1 + \omega$ as $-\omega^2$. This simplifies the base and makes the exponentiation easier. $1 + \omega = -\omega^2$

Step 2: Substitute the simplified base into the given expression. Substitute $1 + \omega = -\omega^2$ into the original equation $(1 + \omega)^7 = A + B\omega$: $(-\omega^2)^7 = A + B\omega$

Step 3: Apply exponent rules to simplify the power. Apply the exponent rules to simplify $(-\omega^2)^7$. We have $(-\omega^2)^7 = (-1)^7(\omega^2)^7$. Since $(-1)^7 = -1$ and $(\omega^2)^7 = \omega^{14}$, the expression becomes: $(-\omega^2)^7 = (-1)^7 (\omega^2)^7 = -1 \cdot \omega^{14} = -\omega^{14}$

Step 4: Reduce the power of $\omega$ using the cube of unity property. We have $-\omega^{14}$. Since $\omega^3 = 1$, we want to find the remainder when 14 is divided by 3. We have $14 = 3 \cdot 4 + 2$, so $\omega^{14} = \omega^{3 \cdot 4 + 2} = (\omega^3)^4 \cdot \omega^2 = 1^4 \cdot \omega^2 = \omega^2$. Therefore, $-\omega^{14} = -\omega^2$. $-\omega^{14} = -\omega^{3 \cdot 4 + 2} = -(\omega^3)^4 \cdot \omega^2 = -1^4 \cdot \omega^2 = -\omega^2$

Step 5: Express the result in the form $A + B\omega$ using the sum of roots property again. We have $-\omega^2$. Using the property $1 + \omega + \omega^2 = 0$, we can rewrite $-\omega^2$ as $1 + \omega$. $-\omega^2 = 1 + \omega$

Step 6: Compare coefficients to find A and B. We have $(1 + \omega)^7 = 1 + \omega$. Comparing this with the given form $A + B\omega$, we have $A = 1$ and $B = 1$. $A + B\omega = 1 + \omega$ $A = 1, \quad B = 1$

Step 7: State the final answer. Thus, the ordered pair $(A, B)$ is $(1, 1)$.

Common Mistakes & Tips

• Sign Errors: Pay close attention to the sign when dealing with negative numbers raised to a power. Remember $(-1)^{\text{odd}} = -1$ and $(-1)^{\text{even}} = 1$.
• Incorrectly Applying $\omega^3=1$: Make sure to divide the exponent by 3 and use the REMAINDER as the new exponent.
• Forgetting the Target Form: Always keep in mind that the final answer must be in the form $A + B\omega$.

Summary

We simplified $(1 + \omega)^7$ using the properties of cube roots of unity. We first used $1 + \omega = -\omega^2$, then applied exponent rules, and finally used $\omega^3 = 1$ to reduce the power of $\omega$. This allowed us to express the result in the form $A + B\omega$ and find the values of $A$ and $B$.

The final answer is \boxed{(1, 1)}, which corresponds to option (A).