<Question>
Simplify and write the result with a rational denominator: $$\sqrt{\sqrt[3]{\sqrt{\frac{1}{729}}}}$$
</Question>

<think>
<Plan_1>
Okay, let's see. I need to simplify this expression: the square root of the cube root of the square root of 1/729, and then write the result with a rational denominator. Hmm, that's a mouthful. Let me break it down step by step.

First, let's write down the original expression to make sure I have it right.
It's √( ∛( √(1/729) ) ). So, starting from the innermost part, we have the square root of 1/729. Then, take the cube root of that result, and finally, take the square root of the cube root result.

Alright, starting with the innermost part: √(1/729). Let's compute that.
The square root of 1 is 1, and the square root of 729... Hmm, what's the square root of 729? Let me think. 27 squared is 729 because 27*27 is 729. Right, so √729 is 27. Therefore, √(1/729) is 1/27.

Next step: take the cube root of 1/27. So, ∛(1/27). The cube root of 1 is 1, and the cube root of 27 is 3, since 3^3 is 27.
Therefore, ∛(1/27) is 1/3.

Now, the outermost operation is the square root of 1/3. So, √(1/3). Let's compute that.

<prompt_1.1> Simplify the expression √(1/3) as per the above plan. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
The square root of 1 is 1, and the square root of 3 is √3. So, √(1/3) is 1/√3. But wait, the problem says to write the result with a rational denominator. Oh, right! We need to rationalize the denominator. To rationalize 1/√3, we multiply numerator and denominator by √3. That gives (1*√3)/(√3*√3) = √3/3.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Let me check each step again to make sure I didn't make a mistake.

Starting with √(1/729):

<prompt_2.1> Verify each step of the simplification process as per the above plan. </prompt_2.1>
-----

Is there another way to approach this problem? Maybe by expressing everything as exponents?

<prompt_2.2> Explore an alternative method using exponents as per the above plan. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
√(1) = 1, √729 = 27, so 1/27. Correct. Then, taking the cube root of 1/27: ∛1 = 1, ∛27 = 3, so 1/3. Then, square root of 1/3 is 1/√3, which becomes √3/3 after rationalizing.
</execution_2.1>

<execution_2.2>
Let's see. The original expression is √( ∛( √(1/729) ) ). Converting roots to exponents, square root is 1/2, cube root is 1/3, and the innermost square root is 1/2. So, starting with 1/729, which is 729^(-1). Then, applying the exponents from the inside out:

First, √(1/729) = (729^(-1))^(1/2) = 729^(-1/2).

Then, taking the cube root of that: (729^(-1/2))^(1/3) = 729^(-1/6).

Finally, taking the square root of that result: (729^(-1/6))^(1/2) = 729^(-1/12).

So, the entire expression simplifies to 729^(-1/12). Now, let's compute 729^(1/12) first. Since 729 is 9^3, which is (3^2)^3 = 3^6. Therefore, 729 = 3^6. So, 729^(1/12) = (3^6)^(1/12) = 3^(6/12) = 3^(1/2) = √3. Therefore, 729^(-1/12) = 1/√3 = √3/3 after rationalizing the denominator.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, but let me confirm if I handled the exponents correctly. Let's think in terms of exponents.

<prompt_3.1> Confirm the simplification using exponent rules as per the above plan. </prompt_3.1>
-----

Based on execution_2.2:
Another check: Let's compute the original expression numerically and see if it matches √3/3.

<prompt_3.2> Perform a numerical verification of the original expression as per the above plan. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
The original expression is √( ∛( √(1/729) ) ). In exponent form, that's ( ( (1/729)^(1/2) )^(1/3) )^(1/2) ). Multiplying the exponents: 1/2 * 1/3 * 1/2 = 1/12. So, (1/729)^(1/12). But 1/729 is 729^(-1), so 729^(-1/12). As before, 729 is 3^6, so 3^6^(-1/12) = 3^(-6/12) = 3^(-1/2) = 1/3^(1/2) = 1/√3 = √3/3.
</execution_3.1>

<execution_3.2>
First, 1/729 is approximately 0.001371742. The square root of that is sqrt(0.001371742) ≈ 0.037. Then, cube root of 0.037 is approximately 0.037^(1/3). Let's see, 0.037 is roughly 1/27, and cube root of 1/27 is 1/3 ≈ 0.333. Then, square root of 0.333 is about 0.577. Now, √3 is approximately 1.732, so √3/3 ≈ 0.577. That matches. So the numerical approximation checks out.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1, execution_2.2, execution_3.2, execution_1.1:
Wait, but let me just go through the steps once again to be thorough.

<prompt_4.1> Review the entire solution process for thoroughness as per the above plan. </prompt_4.1>
-----

Based on execution_3.2:
Therefore, after multiple methods of verification—exponent rules, numerical approximation—it's clear that the simplified form with a rational denominator is √3/3. I think that's solid.

<prompt_4.2> Provide concluding remarks and finalize the answer as per the above plan. </prompt_4.2>
</Plan_4>

<Execution_4>
<execution_4.1>
Starting with 1/729: square root is 1/27. Cube root of 1/27 is 1/3. Square root of 1/3 is 1/√3, which becomes √3/3. All steps seem correct. Also, when rationalizing, we multiply numerator and denominator by √3, which gives (√3)/(√3*√3) = √3/3. Correct.

Is there a possibility that I misapplied the order of operations? Let me check the original expression again. It's sqrt(cuberoot(sqrt(1/729))). So, order is correct: innermost sqrt, then cuberoot, then sqrt again. So the order is correct as I applied.
</execution_4.1>

<execution_4.2>
Just to recap, the key steps were simplifying the innermost radical first, working outward, converting roots to exponents, applying exponent rules, and rationalizing the denominator when necessary. Each step checks out, multiple approaches confirm the same result. So, confident that this is correct.
</execution_4.2>
</Execution_4>

<Final_answer>
Therefore, the simplified expression with a rational denominator is $\boxed{\frac{\sqrt{3}}{3}}$.
</Final_answer>
</think>