Let α, β be two roots of the
equation x² + (20) 1/4 x + (5) 1/2 = 0. Then α⁸ + β⁸ is equal to

Solution

Maths sir · 5 steps

Do not hunt for the roots. Climb from α+β to α⁸+β⁸ by squaring three times.

1. Sum and product. For x² + 20^(1/4)x + √5 = 0, α+β = -20^(1/4) and αβ = √5.

   α²ⁿ+β²ⁿ = (αⁿ+βⁿ)² - 2(αβ)ⁿ

2. Second powers. α²+β² = (α+β)² - 2αβ = √20 - 2√5 = 2√5 - 2√5 = 0.
3. Fourth powers. α⁴+β⁴ = (α²+β²)² - 2(αβ)² = 0 - 2 × 5 = -10.
4. Eighth powers. α⁸+β⁸ = (α⁴+β⁴)² - 2(αβ)⁴ = 100 - 2 × 25 = 50.
5. So. α⁸+β⁸ = 50, which is option (c).

The idea to keep. For a high power sum of two roots, keep doubling the power: square the previous sum and subtract twice the product raised to that power. You never need the roots themselves.

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