Question

1, \(\sqrt{21+12\sqrt{3}}\)

2, \(\sqrt{57-40\sqrt{2}}\)

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

Tính

Answer 1

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

\(=\left|\sqrt{5}+1\right|+\left|\sqrt{5}-1\right|\)

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

\(=2\sqrt{5}\)