Question

 Rút gọn các biểu thức sau:

a) \(\left(\sqrt{3}-\sqrt{2}\right)\sqrt{5+2\sqrt{6}}\)

b) \(\sqrt{24+8\sqrt{5}}+\sqrt{9-4\sqrt{5}}\)

Answer 1

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

\(=\sqrt{15+2.3.\sqrt{6}}\)\(-\sqrt{10+2.2\sqrt{6}}\)

\(=\sqrt{9+2.3\sqrt{6}+6}\)\(-\sqrt{6+2.\sqrt{6}.2+4}\)

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

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

\(=1+0=1\)

Answer 2

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

\(\left(\sqrt{3}-\sqrt{2}\right).\left(\sqrt{3}+\sqrt{2}\right)\)

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

b)  \(\sqrt{24+8\sqrt{5}}+\sqrt{9-4\sqrt{5}}\)

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

=\(2+2\sqrt{5}+\sqrt{5}-2\)\(=3\sqrt{5}\)

Answer 3

a)=\(1\)
b)\(3\sqrt{5}\)

Answer 4

1                3\(\sqrt{5}\)

 

Answer 5

\(a,\left(\sqrt{3}-\sqrt{2}\right)\sqrt{5+2\sqrt{6}}\\ =\left(\sqrt{3}-\sqrt{2}\right).\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}\\ =\left(\sqrt{3}-\sqrt{2}\right).\left(\sqrt{3}+\sqrt{2}\right)\\ =\left(\sqrt{3}\right)^2-\left(\sqrt{2}\right)^2\\ =3-2\\ =1\\ b,\sqrt{24+8\sqrt{5}+\sqrt{9-4\sqrt{5}}}\\ =\sqrt{\left(2+2\sqrt{5}\right)^2+\sqrt{\left(\sqrt{5-2}\right)^2}}\\ =2+2\sqrt{5}-\sqrt{5}-2\\ =3\sqrt{5}\)

Answer 6

a) = 1
b) = \(3\sqrt{5}\)

Answer 7

a) = 1
b) \(3\sqrt{5}\)

Answer 8

a\()\)\((\sqrt{3}-\sqrt{2})\sqrt{5+2\sqrt{6}}\)

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

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

=\((\sqrt{3})^2-(\sqrt{2})^2=3-2=1\)

b\()\)\(\sqrt{24+8\sqrt{5}}+\sqrt{9-4\sqrt{5}}\)

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

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

=3\(\sqrt{5}\)

Answer 9

a) = 1 b) = \(3\sqrt{5}\)

Answer 10

\(a.\left(\sqrt{3}-\sqrt{2}\right)\sqrt{5+2\sqrt{6}}\)

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

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

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

\(=3-2=1\)

\(b.\sqrt{24+8\sqrt{5}}+\sqrt{9-4\sqrt{5}}\)

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

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

\(=3\sqrt{5}\)

Answer 11

a) =1
b) =\(3\sqrt{5}\)

Answer 12

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

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

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

= \(3-2\)= \(1\)

b, \(\sqrt{24+8\sqrt{5}}+\sqrt{9-4\sqrt{5}}\)

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

= \(2+2\sqrt{5}+\sqrt{5}-2\)

= \(3\sqrt{5}\)

Answer 13

 

 

 

Answer 14

\(3\sqrt{5}\)

Answer 15

a) 1

b) 3\(\sqrt{5}\)

Answer 16

a) \(\left(\sqrt{3}-\sqrt{2}\right)\sqrt{5+2\sqrt{6}}\)

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

=3 - 2

= 1

b)\(\sqrt{24+8\sqrt{5}}+\sqrt{9-4\sqrt{5}}\) 

=\(\sqrt{20+8\sqrt{5}+4}+\sqrt{5-4\sqrt{5}+4}\)

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

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

= \(3\sqrt{5}\)

Answer 17

a) = 1
b) = 3\(\sqrt{5}\)

Answer 18

a) 1

b) \(3\sqrt{5}\)

Answer 19

a)1

b)\(3\sqrt{5}\)

Answer 20

a) 1
b) 3\(\sqrt{5}\)

Answer 21

a) 1
b) 3 căn 5

Answer 22

a = 1

b = 3 căn 5

Answer 23

a) 1

b) \(3\sqrt{5}\)

Answer 24

1

\(3\sqrt{5}\)

Answer 25

a) 1

b) 3\(\sqrt{5}\)

Answer 26

a) 1

b) \(3\sqrt{5}\)

Answer 27

a) 1

b) 3\(\sqrt{5}\)

Answer 28

a) 1
b) 3\(\sqrt{5}\)

Answer 29

a)1
b)3\(\sqrt{5}\)

Answer 30

a) 1

b) 3\(\sqrt{5}\)