To solve the problem (assuming it's simplifying (\sqrt{7 + 4\sqrt{3}})), follow these steps:

Step 1: Recognize the expression inside the square root as a perfect square

We need to find two numbers (a) and (b) such that:
((\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} = 7 + 4\sqrt{3})

Comparing terms:

• (a + b = 7)
• (2\sqrt{ab} = 4\sqrt{3} \implies \sqrt{ab} = 2\sqrt{3} \implies ab = 12)

Step 2: Solve for (a) and (b)

The pairs of numbers that multiply to 12 and add to 7 are 4 and 3.

Thus:
((\sqrt{4} + \sqrt{3})^2 = (2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3})

Step 3: Take the square root

(\sqrt{7 + 4\sqrt{3}} = \sqrt{(2 + \sqrt{3})^2} = 2 + \sqrt{3}) (since square roots are non-negative).

Answer: (\boxed{2 + \sqrt{3}})

作者声明:本文包含人工智能生成内容。