Question

Rút gọn các biểu thức sau:

a)\(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)

b)\(\sqrt{m+2\sqrt{m-1}}+\sqrt{m-2\sqrt{m-1}}\)

c)\(\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}\)\(.\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\)\(.\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}\)

Answer 1

b)\(\sqrt{m+2\sqrt{m-1}}+\sqrt{m-2\sqrt{m-1}}\)

\(\Leftrightarrow\sqrt{m-1+2\sqrt{m-1}+1}+\sqrt{m-1-2\sqrt{m-1}+1}\)

\(\Leftrightarrow\sqrt{\left(\sqrt{m-1}+1\right)^2}+\sqrt{\left(\sqrt{m-1}-1\right)^2}\)

\(\Leftrightarrow\sqrt{m-1}+1+\sqrt{m-1}-1\Leftrightarrow2\sqrt{m-1}\)

Answer 2

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\(\sqrt{m-1}\) =\((\sqrt{m-1} -1)^2 \)

Answer 3

a)\(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)

\(\Leftrightarrow\sqrt{13+30\sqrt{2+\sqrt{\left(2\sqrt{2}+1\right)^2}}}\)

\(\Leftrightarrow\sqrt{13+30\sqrt{2+2\sqrt{2}+1}}\)

\(\Leftrightarrow\sqrt{13+30\sqrt{\left(\sqrt{2}+1\right)^2}}\)

\(\Leftrightarrow\sqrt{13+30\sqrt{2}+30}\)

\(\Leftrightarrow\sqrt{43+30\sqrt{2}}\)

\(\Leftrightarrow\sqrt{\left(3\sqrt{2}+5\right)^2}\)

\(\Leftrightarrow3\sqrt{2}+5\)

Answer 4

c)\(\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}.\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}\)

\(\Leftrightarrow\sqrt{\left(2+\sqrt{3}\right)\left(2+\sqrt{2+\sqrt{3}}\right)\left(2+\sqrt{2+\sqrt{2+\sqrt{3}}}\right)\left(2-\sqrt{2+\sqrt{2+\sqrt{3}}}\right)}\)

\(\Leftrightarrow\sqrt{\left(2+\sqrt{3}\right)\left(2+\sqrt{2+\sqrt{3}}\right)\left(4-2-\sqrt{2+\sqrt{3}}\right)}\)

\(\Leftrightarrow\sqrt{\left(2+\sqrt{3}\right)\left(2+\sqrt{2+\sqrt{3}}\right)\left(2-\sqrt{2+\sqrt{3}}\right)}\)

\(\Leftrightarrow\sqrt{\left(2+\sqrt{3}\right)\left(4-2-\sqrt{3}\right)}\)

\(\Leftrightarrow\sqrt{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}\)

\(\Leftrightarrow\sqrt{4-3}\Leftrightarrow1\)