Question

Rút gọn các biểu thức sau:

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

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

\(\sqrt{x-2+2\sqrt{x-3}}-\sqrt{x-3}\)

\(\sqrt{2x-2\sqrt{x^2-4}}+\sqrt{x-2}\)

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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