Question

Rút gọn biểu thức:

\(a,\sqrt{\sqrt{17+12\sqrt{2}}}\)

\(b,\sqrt{4+2\sqrt{3}}-\sqrt{21-12\sqrt{3}}\)

Answer 1

a) \(\sqrt{\sqrt{17+12\sqrt{2}}}\)

\(=\sqrt{\sqrt{\left(2\sqrt{2}\right)^2+2\cdot2\sqrt{2}\cdot3+9}}\)

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

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

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

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

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

Answer 2

b) \(\sqrt{4+2\sqrt{3}}-\sqrt{21-12\sqrt{3}}\)

\(=\sqrt{3+2\sqrt{3}+1}-\sqrt{\left(2\sqrt{2}\right)^2-2\cdot2\sqrt{2}\cdot3+9}\)

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

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

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