Question

The sum of all rational numbers in \( (2 + \sqrt{3})^8 \) is:

• A. 19117
• B. 18817
• C. 18280
• D. 19000

Hint:

In binomial expansions, rational terms are obtained by selecting even powers of irrational numbers.

Answer 1

The Correct Option is C

To determine the sum of the rational terms in the expansion of \( (2 + \sqrt{3})^8 \), apply the binomial theorem:\[(2 + \sqrt{3})^8 = \sum_{k=0}^{8} \binom{8}{k} 2^{8-k} (\sqrt{3})^k\]Rational terms occur when \( k \) is even, as only even powers of \( \sqrt{3} \) result in rational numbers. The applicable even values for \( k \) are \( k = 0, 2, 4, 6, 8 \). The corresponding rational terms are:\[\binom{8}{0} 2^8 (\sqrt{3})^0, \quad \binom{8}{2} 2^6 (\sqrt{3})^2, \quad \binom{8}{4} 2^4 (\sqrt{3})^4, \quad \binom{8}{6} 2^2 (\sqrt{3})^6, \quad \binom{8}{8} 2^0 (\sqrt{3})^8\]Summing these simplified rational terms yields a total of 18280.