Simplify the given mathematical expression: $\sqrt{6}(3+\sqrt{8})$

Step 1 ：The given expression is \(\sqrt{6}(3+\sqrt{8})\)

Step 2 ：Distribute the \(\sqrt{6}\) over the sum of 3 and \(\sqrt{8}\) to get two terms: \(\sqrt{6}*3\) and \(\sqrt{6}*\sqrt{8}\)

Step 3 ：The first term simplifies to \(3\sqrt{6}\)

Step 4 ：The second term simplifies to \(\sqrt{48}\)

Step 5 ：\(\sqrt{48}\) can be further simplified to \(4\sqrt{3}\), since \(48 = 16*3\) and the square root of 16 is 4

Step 6 ：So, the simplified expression is \(3\sqrt{6} + 4\sqrt{3}\)

Step 7 ：Final Answer: \(\boxed{3\sqrt{6} + 4\sqrt{3}}\)