The sum of all rational terms in the expansion of (2+√3)⁸ is :

Solution

Maths sir · 3 steps

A term is rational only when √3 carries an even power, so keep the even r and add them.

1. General term. Tᵣ₊₁ = C(8, r) 2⁸⁻ʳ (√3)ʳ. Since (√3)ʳ = 3^(r/2), the term is rational only when r is even: r = 0, 2, 4, 6, 8.

   r	C(8, r) × 2⁸⁻ʳ × 3^(r/2)	Term
   0	1 × 256 × 1	256
   2	28 × 64 × 3	5376
   4	70 × 16 × 9	10080
   6	28 × 4 × 27	3024
   8	1 × 1 × 81	81

2. Quick check. The rational part equals ((2+√3)⁸ + (2-√3)⁸)/2. Squaring step by step gives (2+√3)⁴ = 97 + 56√3, so the rational part of (2+√3)⁸ is 97² + 3 × 56² = 9409 + 9408 = 18817.
3. So. 256 + 5376 + 10080 + 3024 + 81 = 18817, which is option (b).

The idea to keep. For rational terms of (a+√b)ⁿ, keep the terms where the surd's power is even. Their sum is ((a+√b)ⁿ + (a-√b)ⁿ)/2, which is a quick way to check your answer.

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