Question

\(\sqrt{7+4\sqrt{3}}+\sqrt{7-4\sqrt{3}}\)

Answer 1

\(\sqrt{7+4\sqrt{3}}+\sqrt{7-4\sqrt{3}}\)

\(=\sqrt{7+2\sqrt{12}}+\sqrt{7-2\sqrt{12}}\)

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

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

Answer 2

\(\sqrt{7+4\sqrt{3}}+\sqrt{7-4\sqrt{3}}\)

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

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

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

=4