Question

The sum of all rational terms in the expansion of $ \left( 2 + \sqrt{3} \right)^8 $ is

• A. \(16923\)
• B. \(3763\)
• C. \(33845\)
• D. \(18817\)

Hint:

When expanding expressions of the form \( (a + \sqrt{b})^n \), rational terms are those in which the exponent of \( \sqrt{b} \) is even. Use symmetry by adding \( (a + \sqrt{b})^n \) and \( (a - \sqrt{b})^n \) to quickly eliminate irrational components and double the rational terms.

Answer 1

The Correct Option is D

Let \( S = (2 + \sqrt{3})^8 \). The sum of the rational terms in the binomial expansion is obtained by calculating \( (2 + \sqrt{3})^8 + (2 - \sqrt{3})^8 \). This operation eliminates all irrational terms due to cancellation in the symmetric expansion. Thus, the sum of the rational terms is \( \frac{(2 + \sqrt{3})^8 + (2 - \sqrt{3})^8}{2} \). Alternatively, we can select terms from the binomial expansion where the exponent of \( \sqrt{3} \) is even, ensuring rationality. These terms are: \[ \binom{8}{0}(2)^8 + \binom{8}{2}(2)^6(\sqrt{3})^2 + \binom{8}{4}(2)^4(\sqrt{3})^4 + \binom{8}{6}(2)^2(\sqrt{3})^6 + \binom{8}{8}(\sqrt{3})^8 \] Which simplifies to: \[ 2^8 + 28 \cdot 2^6 \cdot 3 + 70 \cdot 2^4 \cdot 9 + 28 \cdot 2^2 \cdot 27 + 1 \cdot 81 \] And the final sum is: \[ 256 + 5376 + 10080 + 3024 + 81 = 18817 \]