Question

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

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

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

Answer 1

b) \(\left(1+\sqrt{3+\sqrt{5}}\right)\left(1+\sqrt{3-\sqrt{5}}\right)=1+\sqrt{3+\sqrt{5}}+\sqrt{3-\sqrt{5}}+\sqrt{\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)}=\sqrt{3+\sqrt{5}}+\sqrt{3-\sqrt{5}}+3\)Đặt A=\(\sqrt{3+\sqrt{5}}+\sqrt{3-\sqrt{5}}+3\)

⇒A-3=\(\sqrt{3+\sqrt{5}}+\sqrt{3-\sqrt{5}}\)

⇒(A-3)²=\(3+\sqrt{5}+3-\sqrt{5}+2\sqrt{\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)}=6+2\sqrt{4}=10\)

⇒A-3=\(\sqrt{10}\Rightarrow A=\sqrt{10}+3\)

c) Tương tự như câu b)

Chúc bạn học tốt

Answer 2

a) (√(2-√3)+√(2-√3))²=2-√3+2+√3-2√((2-√3)(2+√3))

=4-2√(2²-(√3)²)=4-2√1=4-2=2