Question

Rút gọn: \(\sqrt{5-\sqrt{13+4\sqrt{3}}}+\sqrt{3+\sqrt{13+4\sqrt{3}}}\)

Answer 1

\(\sqrt{5-\sqrt{13+4\sqrt{3}}}+\sqrt{3+\sqrt{13+4\sqrt{3}}}\)

=\(\sqrt{5-\sqrt{13+2.\sqrt{4}\sqrt{3}}}+\sqrt{3+\sqrt{13+2.\sqrt{4}\sqrt{3}}}\)

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

=\(\sqrt{5-\sqrt{\left(\sqrt{12}+1\right)^2}}+\sqrt{3+\sqrt{\left(\sqrt{12}+1\right)^2}}\)

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

=\(\sqrt{4-\sqrt{12}}+\sqrt{4+\sqrt{12}}\)

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

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

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

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