Simplify the numerical expression $\sqrt{\left(\sqrt7-\sqrt5\right)^2}+\sqrt{\left(\sqrt5-\sqrt3\right)^2}-\sqrt{\left(\sqrt3-\sqrt7\right)^2}$.
Luke solved the task as follows: $$\begin{aligned} \sqrt{\left(\sqrt7-\sqrt5\right)^2}+\sqrt{\left(\sqrt5-\sqrt3\right)^2}-\sqrt{\left(\sqrt3-\sqrt7\right)^2}&\stackrel{(1)}=\cr \stackrel{(1)}=\left(\sqrt7-\sqrt5\right)+\left(\sqrt5-\sqrt3\right)-\left(\sqrt3-\sqrt7\right)&\stackrel{(2)}=\cr \stackrel{(2)}=\sqrt7-\sqrt5+\sqrt5-\sqrt3-\sqrt3+\sqrt7&\stackrel{(3)}=\cr \stackrel{(3)}=2\cdot\sqrt7-2\cdot\sqrt3&\stackrel{(4)}=\cr \stackrel{(4)}=2\cdot\left(\sqrt7-\sqrt3\right)& \end{aligned}$$
Is Luke's solution correct? If not, determine where Luke made a mistake in the procedure.
Luke's solution is correct.
The mistake is in equality (1). Luke incorrectly determined the value of some square root.
The mistake is in equality (2). Luke incorrectly removed some parentheses in the expression.
The mistake is in equality (3). Luke incorrectly calculated some values in the expression.
In above presented solution, Luke incorrectly assumed that $\sqrt{a^2}=a$. However, this is true only for $a\geq0$. It is defined that: $$ \sqrt{a^2}=|a|=\left\{\begin{aligned} a\quad \mbox{for}\ a\geq0,\cr -a\quad \mbox{for}\ a<0.\end{aligned}\right. $$
Thus, Luke incorrectly determined the value of the square root $\sqrt{\left(\sqrt3-\sqrt7\right)^2}$. Since $\sqrt3-\sqrt7<0$, it holds $\sqrt{\left( \sqrt3-\sqrt7\right)^2}\neq\sqrt3-\sqrt7$ and $\sqrt{\left(\sqrt3-\sqrt7\right)^2}=\left|\sqrt3-\sqrt7\right|=-\left(\sqrt3-\sqrt7\right)=\sqrt7-\sqrt3$.
Correct solution is: $$\begin{aligned} \sqrt{\left(\sqrt7-\sqrt5\right)^2}+\sqrt{\left(\sqrt5-\sqrt3\right)^2}-\sqrt{\left(\sqrt3-\sqrt7\right)^2}&=\cr =\left(\sqrt7-\sqrt5\right)+\left(\sqrt5-\sqrt3\right)-\left(\sqrt7-\sqrt3\right)&=\cr =\sqrt7-\sqrt5+\sqrt5-\sqrt3-\sqrt7+\sqrt3&=0 \end{aligned}$$