Question

Rút gọn biểu thức:

\(A=\sqrt{16-6\sqrt{7}}+\sqrt{10-2\sqrt{21}}\)

\(B=\sqrt{28+\sqrt{300}}+\sqrt{19-\sqrt{192}}\)

\(C=\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}}\)

Answer 1

\(A=\sqrt{9-6\sqrt{7}+7}+\sqrt{3-2\sqrt{21}+7}\)

\(=\sqrt{\left(3-\sqrt{7}\right)^2}+\sqrt{\left(\sqrt{7}-\sqrt{3}\right)^2}\)

\(=3-\sqrt{7}+\sqrt{7}-\sqrt{3}\)

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

Answer 2

\(B=\sqrt{25+2\sqrt{75}+3}+\sqrt{16-2\sqrt{48}+3}\)

\(=\sqrt{\left(5+\sqrt{3}\right)^2}+\sqrt{\left(4-\sqrt{3}\right)^2}\)

\(=5+\sqrt{3}+4-\sqrt{3}\)

\(=9\)

Answer 3

\(C=\sqrt{\sqrt{5}-\sqrt{3-\sqrt{20-12\sqrt{5}+9}}}\)

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

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

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

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

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