to solve the problem (assuming it's the common expression involving square roots like (\sqrt{7 4\sqrt{3}} \sqrt{7 - 4\sqrt{3}})), follow these steps:
step 1: let (x = \sqrt{7 4\sqrt{3}} \sqrt{7 - 4\sqrt{3}})
square both sides to eliminate the square roots:
[x^2 = \left(\sqrt{7 4\sqrt{3}}\right)^2 \left(\sqrt{7 - 4\sqrt{3}}\right)^2 2 \cdot \sqrt{(7 4\sqrt{3})(7 - 4\sqrt{3})}]
step 2: simplify each term
- (\left(\sqrt{7 4\sqrt{3}}\right)^2 = 7 4\sqrt{3})
- (\left(\sqrt{7 - 4\sqrt{3}}\right)^2 = 7 - 4\sqrt{3})
- the cross product: ((7 4\sqrt{3})(7 - 4\sqrt{3}) = 7^2 - (4\sqrt{3})^2 = 49 - 48 = 1), so (\sqrt{1} = 1)
step 3: compute (x^2)
[x^2 = (7 4\sqrt{3}) (7 - 4\sqrt{3}) 2 \cdot 1 = 14 2 = 16]
step 4: take the positive root
since (x) is a sum of square roots (positive), (x = \sqrt{16} = 4)
answer: (\boxed{4})
作者声明:本文包含人工智能生成内容。
