Question

Rút gọn các biểu thức sau:

a.\(2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}\)

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

c.\(\sqrt{8+\sqrt{40}+\sqrt{20}+\sqrt{8}}\)

d.\(\sqrt{10+\sqrt{24}+\sqrt{20}+\sqrt{8}}\)

d.\(\sqrt{10+\sqrt{24}-\sqrt{40}-\sqrt{60}}\)

Answer 1

a/ \(\sqrt{2}+\sqrt{6}\)

b/ Sửa đề:

\(\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}.\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}=1\)

c/ \(1+\sqrt{2}+\sqrt{5}\)

Answer 2

giải rõ ra hộ mình với

Answer 3

a/ \(2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}\)

\(=2\sqrt{3+\sqrt{5-\sqrt{12+2.2\sqrt{3}+1}}}\)

\(=2\sqrt{3+\sqrt{5-\sqrt{\left(2\sqrt{3}+1\right)^2}}}\)

\(=2\sqrt{3+\sqrt{5-\left(2\sqrt{3}+1\right)}}\)

\(=2\sqrt{3+\sqrt{4-2\sqrt{3}}}\)

\(=2\sqrt{3+\sqrt{3-2\sqrt{3}+1}}\)

\(=2\sqrt{3+\sqrt{\left(\sqrt{3}-1\right)^2}}\)

\(=2\sqrt{3+\left(\sqrt{3}-1\right)}\)

\(=\sqrt{2}\sqrt{4+2\sqrt{3}}\)

\(=\sqrt{2}\sqrt{3+2\sqrt{3}+1}\)

\(=\sqrt{2}\sqrt{\left(\sqrt{3}+1\right)^2}\)

\(=\sqrt{2}\left(\sqrt{3}+1\right)\)

\(=\sqrt{2}+\sqrt{6}\)

Answer 4

b/ \(\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}.\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}\)

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

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

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

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

\(=\sqrt{4-3}=1\)

Answer 5

c/ \(\sqrt{8+\sqrt{40}+\sqrt{20}+\sqrt{8}}\)

\(=\sqrt{5+2+1+2\sqrt{5}\sqrt{2}+2\sqrt{5}+2\sqrt{2}}\)

\(=\sqrt{\left(\sqrt{5}+\sqrt{2}+1\right)^2}=\sqrt{5}+\sqrt{2}+1\)

Answer 6

d/ \(\sqrt{10+\sqrt{24}-\sqrt{40}-\sqrt{60}}\)

\(=\sqrt{2+3+5+2\sqrt{2}\sqrt{3}-2\sqrt{2}\sqrt{5}-2\sqrt{3}\sqrt{5}}\)

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