1.Thực hiện phép tính:
a)\(\sqrt{4+\sqrt{8}}\times\sqrt{2+\sqrt{2+\sqrt{2}}}\times\sqrt{2-\sqrt{2+\sqrt{2}}}\)
b)\(\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+4\sqrt{3}}}}\)
c)
2.Rút gọn: \(\left(\dfrac{5-\sqrt{5}}{\sqrt{5}}-2\right)\times\left(\dfrac{4}{1+\sqrt{5}}+4\right)\)
Có ai không? Giúp mình với!! Nghĩ nát óc mà không ra mấy bài này, mà mai thầy kiểm tra bài rồi huhu
1a) \(\sqrt{4+\sqrt{8}}.\sqrt{2+\sqrt{2+\sqrt{2}}}.\sqrt{2-\sqrt{2+\sqrt{2}}}\)
\(=\sqrt{4+\sqrt{8}}.\sqrt{\left(2+\sqrt{2+\sqrt{2}}\right)\left(\sqrt{2-\sqrt{2+\sqrt{2}}}\right)}\)
\(=\sqrt{4+\sqrt{8}}.\sqrt{4-2-\sqrt{2}}\)
\(=\sqrt{4+\sqrt{8}}.\sqrt{2-\sqrt{2}}=\sqrt{\left(4+\sqrt{8}\right)\left(2-\sqrt{2}\right)}\)
\(=\sqrt{8-4\sqrt{2}-\sqrt{16}+2\sqrt{8}}\)
\(=\sqrt{8-4\sqrt{2}-4+4\sqrt{2}}\)
\(=\sqrt{4}=2\)
1b) \(\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+4\sqrt{3}}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{4+4\sqrt{3}+3}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{\left(2+\sqrt{3}\right)^2}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-20-10\sqrt{3}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{28-10\sqrt{3}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{25-10\sqrt{3}+3}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{\left(5-\sqrt{3}\right)^2}}\)
\(=\sqrt{5\sqrt{3}+25-5\sqrt{3}}\)
\(=\sqrt{25}=5\)
2.
\(\left(\dfrac{5-\sqrt{5}}{\sqrt{5}}-2\right)\left(\dfrac{4}{1+\sqrt{5}}+4\right)\)
\(=\left(\dfrac{5\sqrt{5}-5}{5}-2\right)\left(\dfrac{4\left(1-\sqrt{5}\right)}{1-5}+4\right)\)
\(=\dfrac{5\sqrt{5}-5-10}{5}.\dfrac{4-4\sqrt{5}-16}{-4}\)
\(=\dfrac{5\sqrt{5}-15}{5}.\dfrac{4\sqrt{5}+12}{4}\)
\(=\left(\sqrt{5}-3\right)\left(\sqrt{5}+3\right)\)
\(=5-9=-4\)
