1) \(\sqrt{21+12\sqrt{3}}=\sqrt{3^2+2.3.2\sqrt{3}+\left(2\sqrt{3}\right)^2}=\sqrt{\left(3+2\sqrt{3}\right)^2}\)
\(=\left|3+2\sqrt{3}\right|=3+2\sqrt{3}\)
2) \(\sqrt{57-40\sqrt{2}}=\sqrt{5^2-2.5.4\sqrt{2}+\left(4\sqrt{2}\right)^2}=\sqrt{\left(5-4\sqrt{2}\right)^2}\)
\(=\left|5-4\sqrt{2}\right|=4\sqrt{2}-5\)
3) \(\sqrt{\left(\sqrt{5}+1\right)^2}+\sqrt{\left(\sqrt{5}-1\right)^2}\)
\(=\left|\sqrt{5}+1\right|+\left|\sqrt{5}-1\right|\)
\(=\sqrt{5}+1+\sqrt{5}-1\)
\(=2\sqrt{5}\)