Okay, so I need to simplify the expression 3 divided by the square root of 27. Let me start by recalling what simplifying a radical expression involves. I think the main goal is to rationalize the denominator and simplify the square root as much as possible. Let me break this down step by step.

First, let me look at the denominator, which is sqrt(27). I remember that sqrt(27) can be simplified because 27 is a multiple of a perfect square. The largest perfect square that divides 27 is 9, right? Because 9 times 3 is 27. So, sqrt(27) is the same as sqrt(9*3). Using the property that sqrt(a*b) = sqrt(a)*sqrt(b), this becomes sqrt(9)*sqrt(3). Since sqrt(9) is 3, that simplifies to 3*sqrt(3). So, sqrt(27) simplifies to 3*sqrt(3). 

Now, replacing the denominator in the original expression, we have 3 divided by (3*sqrt(3)). Let me write that out: 3 / (3*sqrt(3)). I notice that the numerator and the denominator both have a factor of 3. So, I can cancel out the 3 in the numerator with the 3 in the denominator. That would leave me with 1 divided by sqrt(3). So, the expression simplifies to 1/sqrt(3). But I think it's usually preferred to rationalize the denominator in such cases. 

To rationalize the denominator, which means to eliminate the square root from the denominator, I need to multiply both the numerator and the denominator by sqrt(3). Let me do that. Multiplying the numerator 1 by sqrt(3) gives sqrt(3), and multiplying the denominator sqrt(3) by sqrt(3) gives sqrt(3)*sqrt(3) = 3. So, the expression becomes sqrt(3)/3. 

Therefore, the simplified form of 3/sqrt(27) is sqrt(3)/3. Let me check my steps again to make sure I didn't make a mistake. Starting with 3/sqrt(27), simplified denominator to 3*sqrt(3), canceled the 3s to get 1/sqrt(3), then rationalized to sqrt(3)/3. Seems correct. Alternatively, I could have approached the problem by combining the constants first. Let's see: 3 divided by sqrt(27) can be written as 3*(1/sqrt(27)). Then, since sqrt(27) is 3*sqrt(3), that becomes 3*(1/(3*sqrt(3))) which is the same as 1/sqrt(3), leading to the same result after rationalizing. 

Alternatively, using exponents, sqrt(27) is 27^(1/2). So, 3 / 27^(1/2) can be written as 3^(1) / 27^(1/2). Since 27 is 3^3, this becomes 3^1 / (3^3)^(1/2) = 3^1 / 3^(3/2) = 3^(1 - 3/2) = 3^(-1/2) = 1/3^(1/2) = 1/sqrt(3). Then, rationalizing gives sqrt(3)/3. Same answer again. 

I think that's thorough. All methods lead to the same simplified form, so I'm confident that sqrt(3)/3 is the correct answer.
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