<Question>
Simplify the following:$$\frac{3}{\sqrt{27}}$$
</Question>

<think>
<Plan_1>
Okay, so I need to simplify the expression 3 divided by the square root of 27. Let me think about how to approach this. First, I remember that simplifying radicals often involves breaking down the denominator into factors that are perfect squares. That way, I can take the square roots of those factors out of the radical. Let me start by looking at the denominator, which is √27.

Hmm, 27 is 3 cubed, right? So 3 times 3 times 3 is 27. But I need to express 27 as a product of a perfect square and another number. The perfect squares I know are 4, 9, 16, 25, etc. Let's see, 9 is a perfect square and 9 times 3 is 27. So I can write √27 as √(9*3).

<prompt_1.1> Simplify the denominator √27 by expressing it as a product of a perfect square and another number, and then separate the square roots. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Then, using the property that √(a*b) = √a * √b, this becomes √9 * √3. Since √9 is 3, that simplifies to 3√3. So √27 is 3√3.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, let me check that again.
If I square 3√3, I get 3^2 * (√3)^2 = 9 * 3 = 27, which matches.

Now, substituting back into the original expression, we have 3 divided by 3√3. So the expression becomes 3/(3√3). 

I notice that the numerator is 3 and the denominator has a 3 as well. So I can cancel out the 3s.
Dividing both numerator and denominator by 3 gives 1/√3. 

But usually, we don't like having a radical in the denominator. Rationalizing the denominator is the next step here. To rationalize, I need to multiply the numerator and the denominator by √3 so that the denominator becomes a rational number.

<prompt_2.1> Rationalize the denominator by multiplying the numerator and denominator by √3. </prompt_2.1>
-----

Alternatively, maybe there's another way to simplify the original expression before rationalizing. Let me think. The original expression is 3/√27. Since 27 is 3^3, so √27 is √(3^3) = 3^(3/2) = 3^1 * 3^(1/2) = 3√3.
So 3 divided by 3√3 is again 1/√3, leading to the same result.

Another check: Convert the original expression to decimal and see if it matches the simplified form. Let's compute 3/√27. √27 is approximately 5.196. So 3 divided by 5.196 is approximately 0.577. Now, 1/√3 is approximately 0.577 as well. So that matches.
</Plan_2>

<Execution_2>
<execution_2.1>
Let's do that:

(1 * √3) / (√3 * √3) = √3 / 3. 

Because the denominator becomes √3 * √3 = 3. So the simplified form is √3 over 3.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, but let me make sure I didn't make any mistakes here. Let me go through each step again.

<prompt_3.1> Review each step of the simplification process to verify the result. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Starting with 3/√27. Simplify √27: √(9*3) = √9 * √3 = 3√3. So denominator is 3√3. Then the expression is 3/(3√3). Cancel the 3s: 1/√3. Then rationalize by multiplying numerator and denominator by √3: (√3)/(√3 * √3) = √3/3. That seems correct.

Alternatively, maybe there's a quicker way. Let me see. If I rationalize the denominator first, but perhaps I can do it in one step. Original expression: 3/√27. Multiply numerator and denominator by √27: (3*√27)/(√27*√27) = (3√27)/27. Simplify denominator: √27*√27 = 27. So numerator is 3√27. Then, 3√27 divided by 27 is (3*√27)/27 = √27/9. Then simplify √27 again: 3√3. So √27/9 = 3√3/9 = √3/3. Same result.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
But usually, the preferred form is to have the denominator rationalized. So √3/3 is the simplified form.

<prompt_4.1> Discuss the standard form and verify the final simplified expression. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Alternatively, sometimes people write 1/√3, but that's not considered simplified because of the radical in the denominator. So √3/3 is the correct simplified form.

Let me check with another method. Let's rationalize the denominator earlier. Original expression: 3/√27. Let's rationalize the denominator by multiplying numerator and denominator by √27:
</execution_4.1>
</Execution_4>

<Final_answer>
Therefore, the simplified form of the given expression is $\boxed{\frac{\sqrt{3}}{3}}$.
</Final_answer>
</think>