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解説
 \( \displaystyle \sqrt{50}+ 2\sqrt{18}-8\sqrt{2}=\sqrt{2\times 5^2}+ 2\sqrt{2\times 3^2}-8\sqrt{2} \)
 \( \displaystyle =5\sqrt{2}+ 6\sqrt{2}-8\sqrt{2}=3\sqrt{2} \)
 \( \displaystyle 3\sqrt{24}- 2\sqrt{6}=3\sqrt{2^2\times 6}-2\sqrt{6} \)
 \( \displaystyle =6\sqrt{6}-2\sqrt{6}=4\sqrt{6} \)
 \( \displaystyle \sqrt{5}- \sqrt{45}=\sqrt{5}-\sqrt{3^2\times 5} \)
 \( \displaystyle =\sqrt{5}-3\sqrt{5}=-2\sqrt{5} \)
 \( \displaystyle \sqrt{75}- 3\sqrt{3}-\sqrt{48}=\sqrt{3\times 5^2}-3\sqrt{3}-\sqrt{3\times 4^2} \)
 \( \displaystyle =5\sqrt{3}-3\sqrt{3}-4\sqrt{3}=-2\sqrt{3} \)
 \( \displaystyle 6\sqrt{2}+ \sqrt{18}=6\sqrt{2}+ \sqrt{2\times 3^2} \)
 \( \displaystyle =6\sqrt{2}+ 3\sqrt{2}=9\sqrt{2} \)
 \( \displaystyle 7\sqrt{6}- \sqrt{24}=7\sqrt{6}- \sqrt{2^2\times 6} \)
 \( \displaystyle =7\sqrt{6}- 2\sqrt{6}=5\sqrt{6} \)
 \( \displaystyle \sqrt{12}+ \sqrt{27}-\sqrt{75}=\sqrt{2^2\times 3}+ \sqrt{3^2 \times 3}- \sqrt{3\times 5^2} \)
 \( \displaystyle =2\sqrt{3}+ 3\sqrt{3}- 5\sqrt{3}=0 \)
 \( \displaystyle \sqrt{20}+ 6\sqrt{5}=\sqrt{2^2\times 5}+ 6\sqrt{5} \)
 \( \displaystyle =2\sqrt{5}+ 6\sqrt{5}=8\sqrt{5} \)
 \( \displaystyle \sqrt{27}-\sqrt{48}+ \sqrt{75}=\sqrt{3^2\times 3}-\sqrt{3\times 4^2}+ \sqrt{3\times 5^2} \)
 \( \displaystyle =3\sqrt{3}-4\sqrt{3}+ 5\sqrt{3}=4\sqrt{3} \)
 \( \displaystyle \sqrt{18}+ \sqrt{2}=\sqrt{2\times 3^2}+ \sqrt{2} \)
 \( \displaystyle =3\sqrt{2}+ \sqrt{2}=4\sqrt{2} \)
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