Multiply and simplify. \[ (3+\sqrt{7})(8+\sqrt{3}) \]

Step 1 ：Given the expression \((3+\sqrt{7})(8+\sqrt{3})\)

Step 2 ：Use the distributive property of multiplication over addition to expand the expression. This property states that for all real numbers a, b, and c, the equation a * (b + c) = a * b + a * c holds.

Step 3 ：Distribute each term in the first parentheses to each term in the second parentheses. This gives us \(3*8 + 3*\sqrt{3} + \sqrt{7}*8 + \sqrt{7}*\sqrt{3}\)

Step 4 ：Simplify the expression by combining like terms.

Step 5 ：\(\boxed{(\sqrt{3} + 8)(\sqrt{7} + 3)}\) is the simplified form of the expression \((3+\sqrt{7})(8+\sqrt{3})\)