Question

Rút gọn các biểu thức sau : 

B=\(\sqrt{x+\sqrt{x^2-1}}-\sqrt{x-\sqrt{x^2-1}}\)  với x>1

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

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

E= \(\sqrt{2x+\sqrt{4x-1}}-\sqrt{2x-\sqrt{4x-1}}\)   với \(\frac{1}{2}\)  < x

Answer 1

\(B=\sqrt{x+\sqrt{x^2-1}}-\sqrt{x-\sqrt{x^2-1}}\)

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

\(B^2=2x-2\sqrt{x^2-x^2+1}\)

\(B^2=2x-2\)

\(\Rightarrow B=\sqrt{2x-2}\)

Answer 2

\(C=\sqrt{x+2\sqrt{x-1}}-\sqrt{x-1}\left(ĐK:x\ge1\right)\)

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

\(C=\sqrt{x-1}+1-\sqrt{x-1}=1\)