Question

Rút gọn các biểu thức sau

a. \(\sqrt{3+2\sqrt{2}}-\sqrt{6-4\sqrt{2}}\)

b. \(\sqrt{9-4\sqrt{5}}-\sqrt{5}\)

c. \(\sqrt{23+8\sqrt{7}}-\sqrt{7}\)

Answer 1

a, \(\sqrt{3+2\sqrt{2}}-\sqrt{6-4\sqrt{2}}\)

= \(\sqrt{2+2\sqrt{2}+1}-\sqrt{4-4\sqrt{2}+2}\)

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

= \(\sqrt{2}\) + 1 - 2 + \(\sqrt{2}\)

= 2\(\sqrt{2}\) - 1

b, \(\sqrt{9-4\sqrt{5}}-\sqrt{5}\)

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

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

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

= - 2

Answer 2

c, \(\sqrt{28+8\sqrt{7}}-\sqrt{7}\)

= \(\sqrt{16+8\sqrt{7}+7}-\sqrt{7}\)

= \(\sqrt{\left(4+\sqrt{7}\right)^2}-\sqrt{7}\)

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

= 4