solution: First, notice that $729=3^6$.  We can begin simplifying from the innermost square root: $$\sqrt{\sqrt[3]{\frac{1}{\sqrt{729}}}}=\sqrt{\sqrt[3]{\frac{1}{27}}}=\sqrt{\frac{1}{3}}=\frac{1}{\sqrt{3}}=\boxed{\frac{\sqrt{3}}{3}}$$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Simplify and write the result with a rational denominator: $$\sqrt{\sqrt[3]{\sqrt{\frac{1}{729}}}}$$
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: \frac{\sqrt{3}}{3}
extra_info.question: Simplify and write the result with a rational denominator: $$\sqrt{\sqrt[3]{\sqrt{\frac{1}{729}}}}$$
extra_info.split: test