Question

Bài 1 : rút gọn

1, \(\sqrt{6+2\sqrt{5}}\)                        2, \(\sqrt{15-6\sqrt{6}}-\sqrt{10-4\sqrt{6}}\)

3. \(\sqrt{31-10\sqrt{6}}-\sqrt{\left(3-2\sqrt{6}\right)^2}\)

Answer 1

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

2.\(\sqrt{10-4\sqrt{6}}=\sqrt{4-4\sqrt{6}+6}=\sqrt{\left(2-\sqrt{6}\right)^2}=\left|2-\sqrt{6}\right|=\sqrt{6}-2\)       \(\sqrt{15-6\sqrt{6}}=\sqrt{9+6\sqrt{6}+6}=\sqrt{\left(3+\sqrt{6}\right)^2}=3+\sqrt{6}\)

                

=>\(\sqrt{15-6\sqrt{6}}-\sqrt{10-4\sqrt{6}}\)=\(3+\sqrt{6}-\sqrt{6}+2\)=5

3. Tương tự bằng :\(8-3\sqrt{6}\)

Answer 2

1) \(\sqrt{6+2\sqrt{5}}\) = \(\sqrt{1+2.1.\sqrt{5}+\sqrt{5}^2}\) =  \(\sqrt{\left(1+\sqrt{5}\right)^2}\)

2) \(\sqrt{15-6\sqrt{6}}-\sqrt{10-4\sqrt{6}}\)

= \(\sqrt{3^2-2.3.\sqrt{6}+\sqrt{6}^2}\) - \(\sqrt{2^2.2.2.\sqrt{6}+\sqrt{6}^2}\)

= \(\sqrt{\left(3+\sqrt{6}\right)^2}\)  - \(\sqrt{\left(2+\sqrt{6}\right)^2}\)

= \(\left|3+\sqrt{6}\right|\) - \(\left|2+\sqrt{6}\right|\)

= 3 + \(\sqrt{6}\) - 2 + \(\sqrt{6}\)

= 1 + 2\(\sqrt{6}\)

3) \(\sqrt{31-10\sqrt{6}}-\sqrt{\left(3-2\sqrt{6}\right)^2}\)

= \(\sqrt{5^2-2.5.\sqrt{6}+\sqrt{6}^2}\) - \(\sqrt{\left(3-2\sqrt{6}\right)^2}\)

= \(\sqrt{\left(5-\sqrt{6}\right)^2}\) - \(\sqrt{\left(3-2\sqrt{6}\right)^2}\)

= \(\left|5-\sqrt{6}\right|\) -  \(\left|3-2\sqrt{6}\right|\)

= 5 - \(\sqrt{6}-3-2\sqrt{6}\)

= 2 - 3\(\sqrt{6}\)

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