If a and b are real numbers such that
( 2 + α )⁴ = a + bα
where α = (- 1 + i√3)/2 then a + b is equal to :

Solution

Maths sir · 3 steps

Spot that α is ω, then keep pushing ω² down to -1-ω until only a + bω is left.

1. Recognise alpha. α = (-1 + i√3)/2 is the complex cube root of unity ω. So ω³ = 1 and 1 + ω + ω² = 0, which gives ω² = -1 - ω.
2. Square once. (2+ω)² = 4 + 4ω + ω² = 4 + 4ω - 1 - ω = 3 + 3ω = 3(1+ω).

   (2+ω)⁴ = 9(1+ω)² = 9(1 + 2ω + ω²) = 9[(1+ω+ω²) + ω] = 9ω

3. So. (2+ω)⁴ = 0 + 9ω, so a = 0 and b = 9. Hence a + b = 9, option (b).

The idea to keep. With powers of ω, reduce ω² to -1-ω and ω³ to 1 at every step. Only then is the answer in the form a + bω, and only then can you read off a and b.

Answer from the NTA final answer key. Solution by PrepWiser.