b)\(\sqrt{m+2\sqrt{m-1}}+\sqrt{m-2\sqrt{m-1}}\)
\(\Leftrightarrow\sqrt{m-1+2\sqrt{m-1}+1}+\sqrt{m-1-2\sqrt{m-1}+1}\)
\(\Leftrightarrow\sqrt{\left(\sqrt{m-1}+1\right)^2}+\sqrt{\left(\sqrt{m-1}-1\right)^2}\)
\(\Leftrightarrow\sqrt{m-1}+1+\sqrt{m-1}-1\Leftrightarrow2\sqrt{m-1}\)
Câu 1 phá từng lớp ra :VD\(9+4\sqrt{2}\) =\((\sqrt{2}+2)^2\)
Câu 2:m+2\(\sqrt{m-1}\) =m-1+1+2\(\sqrt{m-1}\) =\((\sqrt{m-1} -1)^2 \)
a)\(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)
\(\Leftrightarrow\sqrt{13+30\sqrt{2+\sqrt{\left(2\sqrt{2}+1\right)^2}}}\)
\(\Leftrightarrow\sqrt{13+30\sqrt{2+2\sqrt{2}+1}}\)
\(\Leftrightarrow\sqrt{13+30\sqrt{\left(\sqrt{2}+1\right)^2}}\)
\(\Leftrightarrow\sqrt{13+30\sqrt{2}+30}\)
\(\Leftrightarrow\sqrt{43+30\sqrt{2}}\)
\(\Leftrightarrow\sqrt{\left(3\sqrt{2}+5\right)^2}\)
\(\Leftrightarrow3\sqrt{2}+5\)
c)\(\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}.\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}\)
\(\Leftrightarrow\sqrt{\left(2+\sqrt{3}\right)\left(2+\sqrt{2+\sqrt{3}}\right)\left(2+\sqrt{2+\sqrt{2+\sqrt{3}}}\right)\left(2-\sqrt{2+\sqrt{2+\sqrt{3}}}\right)}\)
\(\Leftrightarrow\sqrt{\left(2+\sqrt{3}\right)\left(2+\sqrt{2+\sqrt{3}}\right)\left(4-2-\sqrt{2+\sqrt{3}}\right)}\)
\(\Leftrightarrow\sqrt{\left(2+\sqrt{3}\right)\left(2+\sqrt{2+\sqrt{3}}\right)\left(2-\sqrt{2+\sqrt{3}}\right)}\)
\(\Leftrightarrow\sqrt{\left(2+\sqrt{3}\right)\left(4-2-\sqrt{3}\right)}\)
\(\Leftrightarrow\sqrt{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}\)
\(\Leftrightarrow\sqrt{4-3}\Leftrightarrow1\)