Determine the value of the given mathematical expression: $\sqrt{13^{2}-5^{2}}$

Step 1 ：Given the mathematical expression \(\sqrt{13^{2}-5^{2}}\)

Step 2 ：Recognize this as a difference of squares, which can be simplified using the formula \(\sqrt{a^{2}-b^{2}} = \sqrt{(a+b)(a-b)}\)

Step 3 ：Let \(a=13\) and \(b=5\)

Step 4 ：Simplify the expression inside the square root as \(\sqrt{(13+5)(13-5)}\)

Step 5 ：Calculate the expression inside the square root to get 144

Step 6 ：Take the square root of 144 to get the final answer

Step 7 ：Final Answer: The value of the given mathematical expression is \(\boxed{12}\)