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

<p>To find the sum of all rational terms in the expansion of $(2+\sqrt{3})^8$, we consider the binomial expansion:</p> <p>$ S = { }^8 C_0 (2)^8 + { }^8 C_1 (2)^7 (\sqrt{3}) + \ldots + { }^8 C_8 (\sqrt{3})^8 $</p> <p>We need to identify and sum only the rational terms. In the binomial expansion, a term is rational if the exponent of $\sqrt{3}$ is even.</p> <p>Thus, the rational terms are:</p> <p>$ \begin{aligned} &{ }^8 C_0 (2)^8 + { }^8 C_2 (2)^6 (\sqrt{3})^2 + { }^8 C_4 (2)^4 (\sqrt{3})^4 + \\ &{ }^8 C_6 (2)^2 (\sqrt{3})^6 + { }^8 C_8 (\sqrt{3})^8 \end{aligned} $</p> <p>Evaluating these terms:</p> <p><p>${ }^8 C_0 (2)^8 = 256$</p></p> <p><p>${ }^8 C_2 (2)^6 (3) = { }^8 C_2 \times 64 \times 3 = 1344$</p></p> <p><p>${ }^8 C_4 (2)^4 (3)^2 = { }^8 C_4 \times 16 \times 9 = 3024$</p></p> <p><p>${ }^8 C_6 (2)^2 (3)^3 = { }^8 C_6 \times 4 \times 27 = 4032$</p></p> <p><p>${ }^8 C_8 (3)^4 = 1 \times 81 = 81$</p></p> <p>Summing these rational terms gives:</p> <p>$ 256 + 1344 + 3024 + 4032 + 81 = 18817 $</p> <p>Therefore, the sum of all rational terms in the expansion is $18,817$.</p>