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Ta có: A = \(\sqrt[3]{1+6-5\sqrt{2}}+\sqrt[3]{1+6+5\sqrt{2}}\)
\(=\sqrt[3]{1-3\sqrt{2}+6-2\sqrt{2}}+\sqrt[3]{1+3\sqrt{2}+6+2\sqrt{2}}\)
\(=\sqrt[3]{\left(1-\sqrt{2}\right)^3}+\sqrt[3]{\left(1+\sqrt{2}\right)^3}\)
\(=1-\sqrt{2}+1+\sqrt{2}\)
\(=2\)
Vậy: A luôn là số tự nhiên
Đặt: \(A=\sqrt[3]{7-\sqrt{50}}+\sqrt[3]{7+\sqrt{50}}\)
\(A^3=7-\sqrt{50}+7+\sqrt{50}+3.\left(\sqrt[3]{7-\sqrt{50}}+\sqrt[3]{7+\sqrt{50}}\right).\sqrt[3]{\left(7-\sqrt{50}\right)\left(7+\sqrt{50}\right)}\)\(A^3=14-3A\)
\(A^3+3A-14=0\)
\(A^3-2A^2+2A^2-4A+7A-14=0\)
\(A^2\left(A-2\right)+2A\left(A-2\right)+7\left(A-2\right)=0\)
\(\left(A-2\right)\left(A^2+2A+7\right)=0\)
\(\Rightarrow A-2=0\) ( Do: \(A^2+2A+7>0\) )
\(\Rightarrow A=2\)
\(\Rightarrow A\) \(\in N\)
Cách khác nè :3
\(\sqrt[3]{7-\sqrt{50}}+\sqrt[3]{7+\sqrt{50}}=\sqrt[3]{1-3\sqrt{2}+3.2-2\sqrt{2}}+\sqrt[3]{2\sqrt{2}+3.2+3\sqrt{2}+1}=\sqrt[3]{\left(1-\sqrt{2}\right)^3}+\sqrt[3]{\left(\sqrt{2}+1\right)^3}=1-\sqrt{2}+\sqrt{2}+1=2\)Vậy , biểu thức trên là một số tự nhiên .
\(\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}\)
\(=\sqrt[3]{\left(\sqrt{2}+1\right)^3}+\sqrt[3]{\left(\sqrt{2}-1\right)^3}\)
\(=\sqrt{2}+1+\sqrt{2}-1=2\sqrt{2}\)
a) Ta có: \(\left(7\sqrt{48}+3\sqrt{27}-2\sqrt{12}\right)\cdot\sqrt{3}\)
\(=\left(7\cdot4\sqrt{3}+3\cdot3\sqrt{3}-2\cdot2\sqrt{3}\right)\cdot\sqrt{3}\)
\(=33\sqrt{3}\cdot\sqrt{3}\)
=99
b) Ta có: \(\left(12\sqrt{50}-8\sqrt{200}+7\sqrt{450}\right):\sqrt{10}\)
\(=\left(12\cdot5\sqrt{2}-8\cdot10\sqrt{2}+7\cdot15\sqrt{2}\right):\sqrt{10}\)
\(=\dfrac{85\sqrt{2}}{\sqrt{10}}=\dfrac{85}{\sqrt{5}}=17\sqrt{5}\)
c) Ta có: \(\left(2\sqrt{6}-4\sqrt{3}+5\sqrt{2}-\dfrac{1}{4}\sqrt{8}\right)\cdot3\sqrt{6}\)
\(=\left(2\sqrt{6}-4\sqrt{3}+5\sqrt{2}-\dfrac{1}{4}\cdot2\sqrt{2}\right)\cdot3\sqrt{6}\)
\(=\left(2\sqrt{6}-4\sqrt{3}+3\sqrt{2}\right)\cdot3\sqrt{6}\)
\(=36-36\sqrt{2}+18\sqrt{3}\)
d) Ta có: \(3\sqrt{15\sqrt{50}}+5\sqrt{24\sqrt{8}}-4\sqrt{12\sqrt{32}}\)
\(=3\cdot\sqrt{75\sqrt{2}}+5\cdot\sqrt{48\sqrt{2}}-4\sqrt{48\sqrt{2}}\)
\(=3\cdot5\sqrt{2}\cdot\sqrt{\sqrt{2}}+4\sqrt{3}\sqrt{\sqrt{2}}\)
\(=15\sqrt{\sqrt{8}}+4\sqrt{\sqrt{18}}\)
a,=\(\left(28\sqrt{3}+9\sqrt{3}-4\sqrt{3}\right).\sqrt{3}\)
\(=28.3+9.3-4.3=99\)
b,\(=\left(60\sqrt{2}-80\sqrt{2}+175\sqrt{2}\right):\sqrt{10}\)
\(=155\sqrt{2}:\sqrt{10}=\dfrac{155}{\sqrt{5}}\)
a: \(2\sqrt{8\sqrt{3}}-\sqrt{2\sqrt{3}}-\sqrt{9\sqrt{12}}\)
\(=2\sqrt{4\cdot2\sqrt{3}}-\sqrt{2\sqrt{3}}-\sqrt{9\cdot2\sqrt{3}}\)
\(=4\sqrt{2\sqrt{3}}-\sqrt{2\sqrt{3}}-3\sqrt{2\sqrt{3}}\)
=0
b: \(\sqrt{3}+\sqrt{7-4\sqrt{3}}\)
\(=\sqrt{3}+\sqrt{\left(2-\sqrt{3}\right)^2}\)
\(=\sqrt{3}+\left|2-\sqrt{3}\right|\)
\(=\sqrt{3}+2-\sqrt{3}\)
=2
c: \(\sqrt{\left(\sqrt{7}-4\right)^2}-\sqrt{28}+\sqrt{63}\)
\(=\left|\sqrt{7}-4\right|-2\sqrt{7}+3\sqrt{7}\)
\(=4-\sqrt{7}+\sqrt{7}\)
=4
d: \(\left(15\sqrt{50}+5\sqrt{200}-3\sqrt{450}\right):\sqrt{10}\)
\(=\dfrac{\sqrt{10}\left(15\sqrt{5}+5\sqrt{20}-3\sqrt{45}\right)}{\sqrt{10}}\)
\(=15\sqrt{5}+5\sqrt{20}-3\sqrt{45}\)
\(=15\sqrt{5}+5\cdot2\sqrt{5}-3\cdot3\sqrt{5}\)
\(=16\sqrt{5}\)
e: \(\sqrt{3}-2\sqrt{48}+3\sqrt{75}-4\sqrt{108}\)
\(=\sqrt{3}-2\cdot4\sqrt{3}+3\cdot5\sqrt{3}-4\cdot6\sqrt{3}\)
\(=\sqrt{3}-8\sqrt{3}+15\sqrt{3}-24\sqrt{3}\)
\(=-16\sqrt{3}\)
a: \(A=\left(1-\sqrt{7}\right)\cdot\left(1+\sqrt{7}\right)=1-7=-6\)
b: \(B=3\sqrt{3}+8\sqrt{3}-15\sqrt{3}=-4\sqrt{3}\)
c: \(C=4\sqrt{2}-5\sqrt{2}+3\sqrt{2}=2\sqrt{2}\)
\(A^3=14+3\sqrt[3]{\left(7-\sqrt{50}\right)\left(7+\sqrt{50}\right)}\left(\sqrt[3]{7-\sqrt{50}}+\sqrt[3]{7+\sqrt{50}}\right)\)
\(A^3=14+3\sqrt[3]{49-50}.A\)\(\Leftrightarrow\)\(A^3=14-3A\)
\(\Leftrightarrow\)\(A^3+3A-14=0\)\(\Leftrightarrow\)\(A\left(A^2-4\right)+7\left(A-2\right)=0\)
\(\Leftrightarrow\)\(A\left(A-2\right)\left(A+2\right)+7\left(A-2\right)=0\)
\(\Leftrightarrow\)\(\left(A-2\right)\left(A^2+2A+7\right)=0\)
\(\Leftrightarrow\)\(A=2\) ( do \(A^2+2A+7=\left(A+1\right)^2+6>0\) )