Correct Answer: $2^{\frac{1}{2}}$

Solution : Given: $a = 64$ and $b = 289$
The given expression is $\sqrt{\sqrt{\sqrt{a}+\sqrt{b}}-\sqrt{\sqrt{b}-\sqrt{a}}}$.
Substitute the value of $a = 64$ and $b = 289$ in the above equation, we get,
$=\sqrt{\sqrt{\sqrt{64}+\sqrt{289}}-\sqrt{\sqrt{289} - \sqrt{64}}}$
$=\sqrt{\sqrt{8+17}-\sqrt{17-8}}$
$=\sqrt{\sqrt{25}-\sqrt{9}}$
$=\sqrt{5-3}$
$=2^{\frac{1}{2}}$
Hence, the correct answer is $2^{\frac{1}{2}}$.