\(\sqrt{7+4\sqrt{3}}-\sqrt{4+2\sqrt{3}}\)
\(=\sqrt{2^2+2\cdot2\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{3}\right)^2+2\cdot\sqrt{3}\cdot1+1^2}\)
\(=\sqrt{\left(2+\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{3}+1\right)^2}\)
\(=\left|2+\sqrt{3}\right|-\left|\sqrt{3}+1\right|\)
\(=\left(2+\sqrt{3}\right)-\left(\sqrt{3}+1\right)\)
\(=2+\sqrt{3}-\sqrt{3}-1\)
\(=1\)
\(\sqrt{7+4\sqrt{3}}-\sqrt{4+2\sqrt{3}}\)
\(\sqrt{3+4\sqrt{3}+4}-\sqrt{3+2\sqrt{3}+1}\)
\(=\sqrt{\left(\sqrt{3}\right)^2+2\cdot\sqrt{3}\cdot2+2^2}-\sqrt{\left(\sqrt{3}\right)^2+2\cdot\sqrt{3}\cdot1+1^2}\)
\(=\sqrt{\left(\sqrt{3}+2\right)^2}-\sqrt{\left(\sqrt{3}+1\right)^2}\)
\(=\left|\sqrt{3}+2\right|-\left|\sqrt{3}+1\right|\)
\(=\left(\sqrt{3}+2\right)-\left(\sqrt{3}+1\right)\)
\(=\sqrt{3}+2-\sqrt{3}-1\)
\(=1\)
