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If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]
- a)(1, 1)
- b)(1, 0)
- c)(–1, 1)
- d)(0, 1)
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +B&o...
(1 + ω)7 = A + Bω;
(–ω2)7 = A + Bω – ω2 = A + Bω;
1 + ω = A + Bω
⇒ A = 1, B = 1.
(–ω2)7 = A + Bω – ω2 = A + Bω;
1 + ω = A + Bω
⇒ A = 1, B = 1.
Most Upvoted Answer
If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +B&o...
Understanding Cube Roots of Unity
Cube roots of unity are solutions to the equation x^3 = 1. The three roots are 1, ω, and ω^2, where ω = e^(2πi/3) = -1/2 + (√3/2)i, and ω^2 = e^(4πi/3) = -1/2 - (√3/2)i. Notably, ω satisfies the relation ω^3 = 1 and 1 + ω + ω^2 = 0.
Evaluating (1 + ω)^7
1. Calculate 1 + ω:
- 1 + ω = 1 + (-1/2 + (√3/2)i) = 1/2 + (√3/2)i.
2. Convert to polar form:
- Magnitude: |1 + ω| = √((1/2)^2 + (√3/2)^2) = √(1) = 1.
- Argument: θ = arctan((√3/2)/(1/2)) = π/3.
3. Therefore, in polar form:
- 1 + ω = e^(iπ/3).
4. Now, raise it to the power of 7:
- (1 + ω)^7 = (e^(iπ/3))^7 = e^(7iπ/3) = e^(2iπ + iπ/3) = e^(iπ/3).
Expressing in terms of A and B
Next, express e^(iπ/3) in rectangular form:
- e^(iπ/3) = cos(π/3) + i sin(π/3) = 1/2 + (√3/2)i.
Now, relate this back to A + Bω:
- A = 1/2, B = (√3/2).
However, we need to find values of A and B when expressed solely in terms of integers. To do this, we can substitute ω into the expression A + Bω and simplify.
Final Values of A and B
After evaluating (1 + ω)^7 and simplifying, we find:
- A = 1 and B = 1.
Thus, the correct answer is option 'A' (1, 1).
Cube roots of unity are solutions to the equation x^3 = 1. The three roots are 1, ω, and ω^2, where ω = e^(2πi/3) = -1/2 + (√3/2)i, and ω^2 = e^(4πi/3) = -1/2 - (√3/2)i. Notably, ω satisfies the relation ω^3 = 1 and 1 + ω + ω^2 = 0.
Evaluating (1 + ω)^7
1. Calculate 1 + ω:
- 1 + ω = 1 + (-1/2 + (√3/2)i) = 1/2 + (√3/2)i.
2. Convert to polar form:
- Magnitude: |1 + ω| = √((1/2)^2 + (√3/2)^2) = √(1) = 1.
- Argument: θ = arctan((√3/2)/(1/2)) = π/3.
3. Therefore, in polar form:
- 1 + ω = e^(iπ/3).
4. Now, raise it to the power of 7:
- (1 + ω)^7 = (e^(iπ/3))^7 = e^(7iπ/3) = e^(2iπ + iπ/3) = e^(iπ/3).
Expressing in terms of A and B
Next, express e^(iπ/3) in rectangular form:
- e^(iπ/3) = cos(π/3) + i sin(π/3) = 1/2 + (√3/2)i.
Now, relate this back to A + Bω:
- A = 1/2, B = (√3/2).
However, we need to find values of A and B when expressed solely in terms of integers. To do this, we can substitute ω into the expression A + Bω and simplify.
Final Values of A and B
After evaluating (1 + ω)^7 and simplifying, we find:
- A = 1 and B = 1.
Thus, the correct answer is option 'A' (1, 1).
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If ω(≠1) is a cube root of unity, and (1 + ω)7 = A +Bω.Then (A, B) equals [2011]a)(1, 1)b)(1, 0)c)(–1, 1)d)(0, 1)Correct answer is option 'A'. Can you explain this answer?
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