Question

Rút gọn

\(\sqrt{3}+\sqrt{8-2\sqrt{15}}\) , \(\sqrt{x-1-2\sqrt{x-2}}\)

Answer 1

\(\sqrt{3}+\sqrt{8-2\sqrt{15}}\\ =\sqrt{3}+\sqrt{5-2\sqrt{5\cdot3}+3}\\ =\sqrt{3}+\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}\\ =\sqrt{3}+\sqrt{5}-\sqrt{3}=\sqrt{5}\)

\(\sqrt{x-1-2\sqrt{x-2}}\left(x\ge2\right)\\ =\sqrt{x-2-2\sqrt{x-2}+1}\\ =\sqrt{\left(\sqrt{x-2}-1\right)^2}\\ =\left|\sqrt{x-2}-1\right|\\ =\left[{}\begin{matrix}\sqrt{x-2}-1\left(\sqrt{x-2}\ge1\Leftrightarrow x\ge3\right)\\1-\sqrt{x-2}\left(\sqrt{x-2}< 1\Leftrightarrow2\le x< 3\right)\end{matrix}\right.\)

Chúc bạn học tốt nha.