Question

{ \left( \sqrt{ 4- \sqrt{ 7 \phantom{\tiny{!}}} \phantom{\tiny{!}}} - \sqrt{ 4+ \sqrt{ 7 \phantom{\tiny{!}}} \phantom{\tiny{!}}} \right) }^{ 2 }

Answer 1

$(\sqrt{4-\sqrt{7}}-\sqrt{4+\sqrt{7}})^{2}=4-\sqrt{7}-2\sqrt{(4-\sqrt{7})(4+\sqrt{7})} +4+\sqrt{7}=\\\\=8-2\sqrt{16-7}=8-2\sqrt{9}=8-2*3=\boxed2$

Answer 2

Let's simplify the expression step by step:

1. Start with the innermost square roots:

   • √(7) has no real square root, so it remains as is.

2. Simplify inside the square roots:

   • √(4 - √7) - √(4 + √7)

3. Now, let's find a common denominator for these two square roots, which is (4 - √7)(4 + √7) = (4^2 - 7) = (16 - 7) = 9:

   • √(4 - √7) - √(4 + √7) = √((4 - √7)(4 + √7)) / √9

4. Simplify the numerator:

   • √((4 - √7)(4 + √7)) = √((16 - 7)) = √9 = 3

5. Now, we have:

   • 3 / √9

6. Simplify √9:

   • 3 / √9 = 3 / 3 = 1

7. Finally, square 1:

   • 1^2 = 1

So, the final result is 1.