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$(3+\sqrt{8})^5+(3-\sqrt{8})^5=$
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The correct answer is:
6726
$(3+\sqrt{8})^5+(3-\sqrt{8})^5$
$\begin{aligned} & ={ }^n C_0 3^5+{ }^n C_1 \cdot 3^4 \cdot \sqrt{8}+{ }^n C_2 3^3(\sqrt{8})^2+{ }^n C_3 \cdot 3^2(\sqrt{8})^3 \\ & +{ }^n C_4 \cdot 3(\sqrt{8})+{ }^n C_5 \cdot 3^0(\sqrt{8})^5+{ }^n C_0 3^5+{ }^n C_1 \cdot 3^4(-\sqrt{8})\end{aligned}$
$\begin{aligned} & +{ }^n C_2 \cdot(3)^3(-\sqrt{8})^2+{ }^n C_3(3)^2(-\sqrt{8})^3+{ }^n C_4(3) \cdot(-\sqrt{8})^4 \\ & +{ }^n C_5(3)^0(-\sqrt{8})^5 \\ & =2.3^5+2 \times{ }^5 C_2 3^3 \cdot 8+2 \times{ }^5 C_4 3(64) \\ & =2 \times 243+2 \times 10 \times 27 \times 8+2 \times 5 \times 3 \times 64=6726\end{aligned}$
$\begin{aligned} & ={ }^n C_0 3^5+{ }^n C_1 \cdot 3^4 \cdot \sqrt{8}+{ }^n C_2 3^3(\sqrt{8})^2+{ }^n C_3 \cdot 3^2(\sqrt{8})^3 \\ & +{ }^n C_4 \cdot 3(\sqrt{8})+{ }^n C_5 \cdot 3^0(\sqrt{8})^5+{ }^n C_0 3^5+{ }^n C_1 \cdot 3^4(-\sqrt{8})\end{aligned}$
$\begin{aligned} & +{ }^n C_2 \cdot(3)^3(-\sqrt{8})^2+{ }^n C_3(3)^2(-\sqrt{8})^3+{ }^n C_4(3) \cdot(-\sqrt{8})^4 \\ & +{ }^n C_5(3)^0(-\sqrt{8})^5 \\ & =2.3^5+2 \times{ }^5 C_2 3^3 \cdot 8+2 \times{ }^5 C_4 3(64) \\ & =2 \times 243+2 \times 10 \times 27 \times 8+2 \times 5 \times 3 \times 64=6726\end{aligned}$
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