To solve problems involving rationalizing radicals in the denominator, follow these steps with a common example:

Example Problem: Simplify (\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}})

Step 1: Multiply numerator and denominator by the conjugate of the denominator

The conjugate of (\sqrt{5} - \sqrt{3}) is (\sqrt{5} + \sqrt{3}) (swap the sign between the terms).

[ \frac{(\sqrt{5} + \sqrt{3})(\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})} ]

Step 2: Expand numerator and denominator

• Numerator: Use ((a+b)^2 = a^2 + 2ab + b^2)
  [ (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15} ]

• Denominator: Use ((a-b)(a+b) = a^2 - b^2)
  [ (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 ]

Step 3: Simplify the fraction

[ \frac{8 + 2\sqrt{15}}{2} = 4 + \sqrt{15} ]

Answer: (\boxed{4 + \sqrt{15}}) (adjust based on your specific problem, but this is a common result).

If your problem differs, apply the same logic: conjugate multiplication → expand → simplify. Let me know if you need help with a specific expression!

(\boxed{4 + \sqrt{15}}) (assuming the example problem is correct)
(Note: Replace with your actual result if the problem varies.)
Final Answer
(\boxed{4 + \sqrt{15}})

作者声明:本文包含人工智能生成内容。