Answer
$2$
Work Step by Step
Using the product of the sum and difference of like terms which is given by $(a+b)(a-b)=a^2-b^2,$ the given expression, $
\left( \sqrt{a+1}+\sqrt{a-1} \right)\left( \sqrt{a+1}-\sqrt{a-1} \right)
,$ is equivalent to
\begin{align*}
&
\left( \sqrt{a+1} \right)^2-\left(\sqrt{a-1} \right)^2
\\&=
a+1-(a-1)
\\&=
a+1-a+1
\\&=
2
.\end{align*}
Hence, the simplified form of the given expression is $
2
$.