What is \(\sqrt{17 - 4\sqrt{15}} + \sqrt{8 - 2\sqrt{15}}\) equal to?

Given:

The required expression: \(\sqrt{17 - 4\sqrt{15}} + \sqrt{8 - 2\sqrt{15}}\)

Calculation:

\(\sqrt{17 - 4\sqrt{15}}\) = \(\sqrt{(2\sqrt3)^2 + (\sqrt5)^2 - 2 \times 2\sqrt{3}\times \sqrt5}\) = \(\sqrt{(2\sqrt3 -\sqrt 5)^2}\) = \(2\sqrt3 - \sqrt5\)

\(\sqrt{8 - 2\sqrt{15}}\) = \(\sqrt{(\sqrt{3})^2 + (\sqrt{5})^2 - 2 \times \sqrt{3} \times \sqrt{5}}\) = \(\sqrt{(\sqrt{5} -\sqrt{3})^2 }\) = \(\sqrt{5} - \sqrt{3}\)

Now, adding both terms:

\(2\sqrt3 - \sqrt5\) + \(\sqrt{5} - \sqrt{3}\) = \(\sqrt3\)

Therefore, the final answer is option 1.