Okay, let's see. I need to simplify this expression: the square root of the cube root of the square root of 1/729. Hmm, that sounds a bit complicated, but let me break it down step by step.

First, let me write down the expression to make sure I have it right. The original problem is sqrt(cbrt(sqrt(1/729))). So, starting from the inside, the innermost part is sqrt(1/729). Then, take the cube root of that result, and finally take the square root of the cube root result. Got it.

Let me start with the innermost part: sqrt(1/729). The square root of a fraction is the square root of the numerator over the square root of the denominator. So sqrt(1)/sqrt(729). The square root of 1 is 1, and the square root of 729... Wait, 729 is 27 squared, right? 27 times 27 is 729. So sqrt(729) is 27. Therefore, sqrt(1/729) is 1/27. Okay, that part is straightforward.

Now, moving outward, the next step is to take the cube root of 1/27. So cbrt(1/27). Again, the cube root of a fraction is the cube root of the numerator over the cube root of the denominator. The cube root of 1 is 1, and the cube root of 27 is 3 because 3^3 = 27. So cbrt(1/27) is 1/3. So far, so good.

Now, the outermost operation is the square root of 1/3. So sqrt(1/3). Similar to before, sqrt(1)/sqrt(3) is 1/sqrt(3). But the problem says to write the result with a rational denominator. Rationalizing the denominator means getting rid of the square root in the denominator. To do that, I can multiply both the numerator and the denominator by sqrt(3). So 1/sqrt(3) becomes (1*sqrt(3))/(sqrt(3)*sqrt(3)) which is sqrt(3)/3. So sqrt(1/3) simplifies to sqrt(3)/3.

Putting it all together: the original expression simplifies to sqrt(3)/3. Let me check if each step is correct.

Starting with sqrt(1/729) = 1/27. Then cbrt(1/27) = 1/3. Then sqrt(1/3) = sqrt(3)/3. Yes, that seems right. Let me verify each operation again.

First, sqrt(1/729): 1/729 is (1/27)^2 because 27^2 = 729. So sqrt((1/27)^2) is 1/27. Correct. Then, cube root of 1/27: 27 is 3^3, so 1/27 is (1/3)^3. Cube root of that is 1/3. Then sqrt(1/3) is 1/sqrt(3), which rationalizes to sqrt(3)/3. All steps check out.

Alternatively, maybe there's a way to combine all the roots into a single exponent. Let's see. The original expression is sqrt(cbrt(sqrt(1/729))). Let's write each root as an exponent.

sqrt(x) is x^(1/2), cbrt(x) is x^(1/3), and so on. So starting from the inside: sqrt(1/729) = (1/729)^(1/2). Then cbrt of that is [(1/729)^(1/2)]^(1/3) = (1/729)^(1/2 * 1/3) = (1/729)^(1/6). Then the sqrt of that is [(1/729)^(1/6)]^(1/2) = (1/729)^(1/6 * 1/2) = (1/729)^(1/12). 

But 1/729 is 3^(-6) because 729 is 3^6 (since 3^6 = 729). So (1/729)^(1/12) is (3^(-6))^(1/12) = 3^(-6/12) = 3^(-1/2) = 1/(3^(1/2)) = 1/sqrt(3). Which is the same as sqrt(3)/3. So that confirms the previous result.

Therefore, the simplified expression with a rational denominator is sqrt(3)/3. I think that's the correct answer. Let me just verify once more with another approach.

Alternatively, let's compute the original expression numerically. Let's compute each step numerically to see if we get the same result.

First, 1/729 is approximately 0.00137174211. Then sqrt(0.00137174211) is approximately 0.037037037. Then cube root of 0.037037037. Let's compute that. The cube root of 0.037037037. Since 0.037037037 is approximately 1/27, and cube root of 1/27 is 1/3, which is approximately 0.333333333. Then sqrt(0.333333333) is approximately 0.577350269. Now, sqrt(3)/3 is approximately 1.732050807/3 = 0.577350269, which matches the numerical value. So the decimal approximation checks out. Therefore, the algebraic result sqrt(3)/3 is correct.

Therefore, after simplifying and rationalizing the denominator, the final answer is sqrt(3)/3.
Therefore, the simplified expression with a rational denominator is $\boxed{\frac{\sqrt{3}}{3}}$.
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