Question:

<p>The expansion of <span class="math-tex">$(2+\sqrt{3})^{8}$</span> is given by:<br /> <span class="math-tex">$\sum_{k=0}^{8} C_{k} 2^{8-k}(\sqrt{3})^{k}$</span><br /> A term is rational only when the exponent of <span class="math-tex">$\sqrt{3}$</span> is even (i.e., <span class="math-tex">$k=0,2,4,6,8$</span> ):<br /> <span class="math-tex">${ }^{8} C_{0} 2^{8}(\sqrt{3})^{0}+{ }^{8} C_{2} 2^{6}(\sqrt{3})^{2}+{ }^{8} C_{4} 2^{4}(\sqrt{3})^{4}$</span><br /> <span class="math-tex">$+{ }^{8} C_{6} 2^{2}(\sqrt{3})^{6}+{ }^{8} C_{8} 2^{0}(\sqrt{3})^{8}$</span><br /> For <span class="math-tex">$k=0$</span> :<br /> <span class="math-tex">${ }^{8} C_{0} 2^{8}=1 \times 256=256$</span><br /> For <span class="math-tex">$k=2$</span> :<br /> <span class="math-tex">${ }^{8} C_{2} 2^{6} \times 3=28 \times 64 \times 3=5376$</span><br /> For <span class="math-tex">$k=4$</span> :<br /> <span class="math-tex">${ }^{8} C_{4} 2^{4} \times 9=70 \times 16 \times 9=10080$</span><br /> For <span class="math-tex">$k=6$</span> :<br /> <span class="math-tex">${ }^{8} C_{6} 2^{2} \times 27=28 \times 4 \times 27=3024$</span><br /> For <span class="math-tex">$k=8:{ }^{8} C_{8} \times 81=1 \times 81=81$</span><br /> Sum the Rational Terms:<br /> <span class="math-tex">$256+5376+10080+3024+81=18817$</span></p>