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

Giải:

1) \(\sqrt{21+12\sqrt{3}}\)

\(=\sqrt{12+9+12\sqrt{3}}\)

\(=\sqrt{12+12\sqrt{3}+9}\)

\(=\sqrt{\left(2\sqrt{3}\right)^2+2.2\sqrt{3}.3+3^2}\)

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

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

Vậy ...

2) \(\sqrt{57-40\sqrt{2}}\)

\(=\sqrt{32+25-40\sqrt{2}}\)

\(=\sqrt{32-40\sqrt{2}+25}\)

\(=\sqrt{\left(4\sqrt{2}\right)^2-2.4\sqrt{2}.5+5^2}\)

\(=\sqrt{\left(4\sqrt{2}-5\right)^2}\)

\(=4\sqrt{2}-5\)

Vậy ...

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

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

\(=2\sqrt{5}\)

Vậy ...