Tübitak Lise 1. Aşama 2006 Soru 23

[ERhan ERdoğan]

$\left ( 1+\dfrac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4} \right )^{10}$ sayısını aşmayan en büyük tam sayı kaçtır?

$
\textbf{a)}\ 2
\qquad\textbf{b)}\ 10
\qquad\textbf{c)}\ 21
\qquad\textbf{d)}\ 32
\qquad\textbf{e)}\ 36
$

[geo]

Yanıt: $\boxed{D}$

$\begin{array}{rcl}
\left ( 1+\dfrac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4} \right )^{10} &=& \left ( 1+\dfrac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{4}+\sqrt 6 + \sqrt 8} \right )^{10} \\
&=& \left ( 1+\dfrac{1}{1+\sqrt 2} \right )^{10} \\
&=& \left ( \dfrac{2+\sqrt 2}{1+\sqrt 2} \right )^{10} \\
&=& (\sqrt 2)^{10} \\
&=& 32
\end{array}$