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\(\sqrt{242}.\sqrt{26}.\sqrt{130}.\sqrt{0,9}-\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)\)
\(=\sqrt{121}.\sqrt{2}.\sqrt{2}.\sqrt{13}.\sqrt{13}.\sqrt{10}.\sqrt{0,9}-\left(2-1\right)\)
\(=11.2.13.\sqrt{9}-1=286.3-1=857\)
\(\frac{3-\sqrt{6}}{\sqrt{12}-\sqrt{8}}-\frac{\sqrt{15}-\sqrt{5}}{2\sqrt{12}-4}+\frac{\sqrt{17-4\sqrt{15}}}{4}\)
\(=\frac{\sqrt{3}\left(\sqrt{3}-\sqrt{2}\right)}{2\left(\sqrt{3}-\sqrt{2}\right)}-\frac{\sqrt{5}\left(\sqrt{3}-1\right)}{4\left(\sqrt{3}-1\right)}+\frac{\sqrt{\left(2\sqrt{3}-\sqrt{5}\right)^2}}{4}\)
\(=\frac{\sqrt{3}}{2}-\frac{\sqrt{5}}{4}+\frac{2\sqrt{3}-\sqrt{5}}{4}\)
\(=\sqrt{3}-\frac{\sqrt{5}}{4}\)
\(a.\left(2\sqrt{2}-\sqrt{3}\right)^2=8-4\sqrt{6}+3=11-4\sqrt{6}\)
\(b.\left(1+\sqrt{3}-\sqrt{2}\right)\left(1+\sqrt{3}+\sqrt{2}\right)=\left(1+\sqrt{3}\right)^2-2=4+2\sqrt{3}-2=2+2\sqrt{3}\) \(c.\left(\sqrt{3-\sqrt{5}}+\sqrt{3+\sqrt{5}}\right)^2=3-\sqrt{5}+3+\sqrt{5}+2\sqrt{9-5}=6+4=10\) \(d.\left(\sqrt{\sqrt{11}+\sqrt{7}}-\sqrt{\sqrt{11}-\sqrt{7}}\right)^2=\sqrt{11}+\sqrt{7}+\sqrt{11}-\sqrt{7}-2\sqrt{11-7}=2\sqrt{11}-4\) \(e.\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}=\dfrac{\sqrt{3+2\sqrt{3}+1}-\sqrt{3-2\sqrt{3}+1}}{\sqrt{2}}=\dfrac{\sqrt{3}+1-\sqrt{3}+1}{\sqrt{2}}=\sqrt{2}\) \(f.\sqrt{21-12\sqrt{3}}-\sqrt{3}=\sqrt{12-2.2\sqrt{3}.3+9}-\sqrt{3}=2\sqrt{3}-3-\sqrt{3}=\sqrt{3}-3\)
\(g.\left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{3}-2\right)\sqrt{\sqrt{3}+2}=\left(\sqrt{3}+1\right)\left(\sqrt{3}-2\right)\sqrt{3+2\sqrt{3}+1}=\left(\sqrt{3}+1\right)^2\left(\sqrt{3}-2\right)=\left(4+2\sqrt{3}\right)\left(\sqrt{3}-2\right)=2\left(2+\sqrt{3}\right)\left(\sqrt{3}-2\right)=2\left(3-4\right)=-2\)
\(h.\sqrt{6-2\sqrt{\sqrt{2}+\sqrt{12}+\sqrt{18-\sqrt{128}}}}=\sqrt{6-2\sqrt{\sqrt{2}+2\sqrt{3}+\sqrt{16-2.4\sqrt{2}+2}}}=\sqrt{6-2\sqrt{\sqrt{2}+2\sqrt{3}+4-\sqrt{2}}}=\sqrt{6-2\sqrt{3+2\sqrt{3}+1}}=\sqrt{6-2\left(\sqrt{3}+1\right)}=\sqrt{3-2\sqrt{3}+1}=\sqrt{3}-1\)
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d/ \(x=\sqrt[3]{3+\sqrt{9+\frac{125}{27}}}-\sqrt[3]{-3+\sqrt{9+\frac{125}{27}}}\)
\(\Leftrightarrow x^3=3+\sqrt{9+\frac{125}{27}}+3-\sqrt{9+\frac{125}{27}}-3\left(\sqrt[3]{3+\sqrt{9+\frac{125}{27}}}-\sqrt[3]{-3+\sqrt{9+\frac{125}{27}}}\right)\sqrt[3]{3+\sqrt{9+\frac{125}{27}}}.\sqrt[3]{-3+\sqrt{9+\frac{125}{27}}}\)
\(\Leftrightarrow x^3=6-3x\sqrt[3]{9-9-\frac{125}{27}}\)
\(\Leftrightarrow x^3=6-5x\)
\(\Leftrightarrow\left(x-1\right)\left(x^2+x+6\right)=0\)
\(\Leftrightarrow x=1\)
c/
\(\left(\sqrt{3}-1\right)\sqrt{6+2\sqrt{2}\sqrt{3-\sqrt{\sqrt{2}+\sqrt{12}+\sqrt{18-\sqrt{128}}}}}\)
\(=\left(\sqrt{3}-1\right)\sqrt{6+2\sqrt{2}\sqrt{3-\sqrt{\sqrt{2}+\sqrt{12}+\sqrt{\left(4-\sqrt{2}\right)^2}}}}\)
\(=\left(\sqrt{3}-1\right)\sqrt{6+2\sqrt{2}\sqrt{3-\sqrt{\sqrt{12}+4}}}\)
\(=\left(\sqrt{3}-1\right)\sqrt{6+2\sqrt{2}\sqrt{3-\sqrt{\left(\sqrt{3}+1\right)^2}}}\)
\(=\left(\sqrt{3}-1\right)\sqrt{6+2\sqrt{2}\sqrt{2-\sqrt{3}}}\)
\(=\left(\sqrt{3}-1\right)\sqrt{6+2\sqrt{4-2\sqrt{3}}}\)
\(=\left(\sqrt{3}-1\right)\sqrt{6+2\sqrt{\left(\sqrt{3}-1\right)^2}}\)
\(=\left(\sqrt{3}-1\right)\sqrt{4+2\sqrt{3}}\)
\(=\left(\sqrt{3}-1\right)\sqrt{\left(\sqrt{3}+1\right)^2}\)
\(=\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)\)
\(=3-1=2\)
\(D=\left(\sqrt{3}-1\right)\cdot\sqrt{6+2\sqrt{2}\cdot\sqrt{\sqrt{2}+\sqrt{12}+\sqrt{18}-\sqrt{128}}}\\ D=\left(\sqrt{3}-1\right)\cdot\sqrt{6+2\sqrt{2}\cdot\sqrt{\sqrt{2}+2\sqrt{2}+3\sqrt{2}-8\sqrt{2}}}\\ D=\left(\sqrt{3}-1\right)\cdot\sqrt{6+2\sqrt{2}\cdot\left(-2\sqrt{2}\right)}\\ D=\left(\sqrt{3}-1\right)\cdot\sqrt{6+\sqrt{12}\cdot\left(-\sqrt{12}\right)}\\ D=\left(\sqrt{3}-1\right)\cdot\sqrt{6+\left(-12\right)}\\ D=\left(\sqrt{3}-1\right)\cdot\sqrt{6}\\ D=\sqrt{18}-\sqrt{6}\)