\(\left(7+\sqrt{14}\right).\sqrt{9-2\sqrt{14}}\)
\(\Leftrightarrow\sqrt{7}\left(\sqrt{7}+\sqrt{2}\right).\sqrt{\left(\sqrt{7}-\sqrt{2}\right)^2}\)
\(\Leftrightarrow\sqrt{7}\left(\sqrt{7}+\sqrt{2}\right).\left(\sqrt{7}-\sqrt{2}\right)\)
\(\Leftrightarrow\sqrt{7}\left(7-2\right)\)
\(\Leftrightarrow5\sqrt{7}\)
\(\sqrt{2}.\sqrt{7-3\sqrt{5}}\)
\(\Leftrightarrow\sqrt{2\left(7-3\sqrt{5}\right)}\)
\(\Leftrightarrow\sqrt{14-6\sqrt{5}}\)
\(\Leftrightarrow\sqrt{\left(3-\sqrt{5}\right)^2}\)
\(\Leftrightarrow3-\sqrt{5}\)
a)\(\sqrt{2}.\sqrt{7-3\sqrt{5}}=\sqrt{14-2.3\sqrt{5}}=\sqrt{\left(3-\sqrt{5}\right)^2}=3-\sqrt{5}\)
b)\(\sqrt{\dfrac{59}{25}+\dfrac{6}{5}\sqrt{2}}=\sqrt{\dfrac{9}{25}+2.\dfrac{3}{5}.\sqrt{2}+2}=\sqrt{\left(\dfrac{3}{5}+\sqrt{2}\right)^2}=\dfrac{3}{5}+\sqrt{2}\)
c)\(2.\left(\sqrt{10}-\sqrt{2}\right).\sqrt{4+\sqrt{6-2\sqrt{5}}}\)
\(=2\sqrt{2}.\left(\sqrt{5}-1\right).\sqrt{4+\sqrt{\left(\sqrt{5}-1\right)^2}}\)
\(=2\sqrt{2}\left(\sqrt{5}-1\right)\sqrt{3+\sqrt{5}}\)
\(=2\left(\sqrt{5}-1\right)\sqrt{6+2\sqrt{5}}\)
\(=2\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)\)
\(=2\left(5-1\right)=2.4=8\)
d)\(\left(7+\sqrt{14}\right).\sqrt{9-2\sqrt{14}}\)
\(=\left(7+\sqrt{14}\right).\sqrt{\left(\sqrt{7}-\sqrt{2}\right)^2}\)
\(=\sqrt{7}.\left(\sqrt{7}+\sqrt{2}\right).\left(\sqrt{7}-\sqrt{2}\right)\)
\(=\sqrt{7}.\left(7-2\right)=5\sqrt{7}\)
