Question

Упростить выражение:

Answer 1

$\displaystyle\bf\\1) \ \ \sqrt{a+2\sqrt{a+1}+2 } +\sqrt{a-2\sqrt{a+1}+2 } =\\\\\\=\sqrt{a+1+2\sqrt{a+1}+1 } +\sqrt{a+1-2\sqrt{a+1}+1 } =\\\\\\=\sqrt{\Big(\sqrt{a+1}\Big)^{2} +2\sqrt{a+1}+1 } +\sqrt{\Big(\sqrt{a+1}\Big)^{2} -2\sqrt{a+1}+1 } =\\\\\\=\sqrt{\Big(\sqrt{a+1} +1\Big)^{2} } +\sqrt{\Big(\sqrt{a+1} -1\Big)^{2} } =\Big|\sqrt{a+1} +1\Big|+\Big|\sqrt{a+1} -1\Big|=\\\\\\=\sqrt{a+1} +1+1-\sqrt{a+1} =2 \ \ \ ; \ -1\leq a\leq 0\\\\2)$

$\displaystyle\bf\\a > 0 \ \ \ \ \Rightarrow \ \ \Big|\sqrt{a+1} +1\Big|+\Big|\sqrt{a+1} -1\Big|=\sqrt{a+1} +1+\sqrt{a+1} -1=2\sqrt{a+1}$