<Question>
If we write $\sqrt{2}+\frac{1}{\sqrt{2}} + \sqrt{3} + \frac{1}{\sqrt{3}}$ in the form $\dfrac{a\sqrt{2} + b\sqrt{3}}{c}$ such that $a$, $b$, and $c$ are positive integers and $c$ is as small as possible, then what is $a+b+c$?
</Question>

<think>
<Plan_1>
Okay, so I need to take this expression: √2 + 1/√2 + √3 + 1/√3, and rewrite it in the form (a√2 + b√3)/c, where a, b, and c are positive integers, and c is as small as possible. Then find a + b + c. Hmm, let's break this down step by step.

<prompt_1.1> Understand the problem and plan the approach to combine the terms and simplify the expression. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, I notice that the expression has four terms: two square roots (√2 and √3) and their reciprocals (1/√2 and 1/√3). The goal is to combine all these terms into a single fraction with a numerator that's a combination of √2 and √3 terms, and a denominator that's a positive integer, minimized. 

So, maybe I should start by simplifying each of the reciprocal terms. Let's recall that 1/√n can be rationalized by multiplying numerator and denominator by √n. So, 1/√2 is √2/2, and 1/√3 is √3/3. Let me write that out:

Original expression: √2 + 1/√2 + √3 + 1/√3

After rationalizing the denominators:
√2 + (√2)/2 + √3 + (√3)/3
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, let's combine like terms. The terms with √2 are √2 and (√2)/2, and the terms with √3 are √3 and (√3)/3.

<prompt_2.1> Combine the simplified terms to form a single expression. </prompt_2.1>
-----

But let me check if there's another way to combine the terms that might lead to a smaller denominator. Alternatively, maybe if we pair √2 terms and √3 terms differently? Let's see.

<prompt_2.2> Explore alternative methods to combine the terms and find a smaller denominator. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
Let's handle the √2 terms first. 

√2 is the same as 2√2/2, right? Because multiplying numerator and denominator by 2. So, 2√2/2 + √2/2 = (2√2 + √2)/2 = 3√2/2. 

Similarly, for the √3 terms: √3 is the same as 3√3/3. So, 3√3/3 + √3/3 = (3√3 + √3)/3 = 4√3/3.
</execution_2.1>

<execution_2.2>
Suppose instead of rationalizing each term individually, maybe we can combine the terms first. Let's see:

Original expression: √2 + 1/√2 + √3 + 1/√3

If we group √2 and 1/√2 together, and √3 and 1/√3 together:

(√2 + 1/√2) + (√3 + 1/√3)

Which is the same as:

(√2 + √2/2) + (√3 + √3/3) = (3√2/2) + (4√3/3)
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
So now the entire expression becomes:
3√2/2 + 4√3/3

But we need to combine these two terms into a single fraction. To do that, we need a common denominator. The denominators here are 2 and 3.
The least common denominator (LCD) of 2 and 3 is 6. So, let's convert each fraction to have denominator 6.

<prompt_3.1> Find a common denominator and combine the fractions. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
For 3√2/2: Multiply numerator and denominator by 3: (3√2 * 3)/(2 * 3) = 9√2/6

For 4√3/3: Multiply numerator and denominator by 2: (4√3 * 2)/(3 * 2) = 8√3/6

Now, adding these two fractions together: 9√2/6 + 8√3/6 = (9√2 + 8√3)/6
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
So, according to this, a = 9, b = 8, c = 6. Then, a + b + c = 9 + 8 + 6 = 23.

But wait, the problem states that c should be as small as possible. Is 6 the smallest possible denominator here?

Let me check if there's a way to simplify this further or if there's a common factor in the numerator that can be factored out to reduce the denominator.

<prompt_4.1> Verify if the current form is the simplest and if 6 is the smallest possible denominator. </prompt_4.1>
-----

Based on execution_3.1, execution_2.2:
So, combining these gives 3√2/2 + 4√3/3, which we already determined leads to (9√2 + 8√3)/6. Since we can't combine these terms further because they have different radicals, this seems to be the simplest form.

But let me check again if there's a way to represent this with a smaller denominator. Suppose, for some reason, we didn't combine the terms first. Let's try another approach.

<prompt_4.2> Confirm the final form and check for any overlooked simplifications. </prompt_4.2>
</Plan_4>

<Execution_4>
<execution_4.1>
Looking at the numerator: 9√2 + 8√3. Since √2 and √3 are unlike terms, we can't combine them further. Therefore, the numerator is already in its simplest form. The denominator is 6. 

Is there a common factor in the numerator that could be factored out to reduce the denominator? Let's see. The coefficients are 9 and 8. The greatest common divisor (GCD) of 9 and 8 is 1. So, there's no common factor to factor out. Therefore, the denominator 6 is already the smallest possible.
</execution_4.1>

<execution_4.2>
Original expression: √2 + 1/√2 + √3 + 1/√3

If we don't combine the terms first, maybe we can write all terms over a common denominator. Let's see. The denominators we have are 1, √2, 1, √3. Wait, but denominators with radicals are not ideal. So, rationalizing each term first is better. As we did before.

Alternatively, if we consider writing all terms with denominators. Let's see:

√2 is the same as √2/1, 1/√2 is √2/2, √3 is √3/1, and 1/√3 is √3/3. So, all terms can be written as:

√2/1 + √2/2 + √3/1 + √3/3

Now, to combine all these terms into a single fraction, we need a common denominator. The denominators here are 1, 2, 1, 3. The LCD of 1, 2, 1, 3 is 6. So, converting each term:

√2/1 = 6√2/6

√2/2 = 3√2/6

√3/1 = 6√3/6

√3/3 = 2√3/6

Adding all these up:

6√2/6 + 3√2/6 + 6√3/6 + 2√3/6 = (6√2 + 3√2 + 6√3 + 2√3)/6

Combine like terms:

(6√2 + 3√2) = 9√2

(6√3 + 2√3) = 8√3

So, total numerator: 9√2 + 8√3 over denominator 6, same as before. So, that confirms the earlier result.
</execution_4.2>
</Execution_4>

<Plan_5>
Based on execution_4.1:
But let me double-check my steps to ensure I didn't make a mistake.

<prompt_5.1> Verify the entire solution process to ensure no mistakes were made. </prompt_5.1>

But let me check once more if there's a way to simplify further. For instance, if the numerator can be expressed as a multiple of a common term, but since √2 and √3 are different radicals, they can't be combined. Therefore, the numerator is indeed in its simplest form, and the denominator is 6, which can't be reduced further.

<prompt_5.2> Perform a final verification to confirm the solution is correct. </prompt_5.2>
</Plan_5>

<Execution_5>
<execution_5.1>
Original expression: √2 + 1/√2 + √3 + 1/√3

Rationalizing 1/√2 gives √2/2, and 1/√3 gives √3/3. Then:

√2 + √2/2 + √3 + √3/3

Combining like terms:

For √2: 1 + 1/2 = 3/2, so 3√2/2

For √3: 1 + 1/3 = 4/3, so 4√3/3

Then adding these two results: 3√2/2 + 4√3/3

Convert to common denominator 6:

3√2/2 = 9√2/6

4√3/3 = 8√3/6

Adding them gives (9√2 + 8√3)/6

Yes, that seems correct. So a = 9, b = 8, c = 6. Sum is 23.
</execution_5.1>

<execution_5.2>
Therefore, the answer should be 9 + 8 + 6 = 23. 

Wait, but hold on a second. Let me verify each step again because sometimes when you rationalize and combine terms, you might have miscalculations. Let's redo each step quickly:

Original expression:

√2 + 1/√2 + √3 + 1/√3

Rationalize 1/√2 = √2/2

Rationalize 1/√3 = √3/3

So, expression becomes:

√2 + √2/2 + √3 + √3/3

Combine √2 terms:

√2 is equal to 2√2/2, so 2√2/2 + √2/2 = 3√2/2

Combine √3 terms:

√3 is equal to 3√3/3, so 3√3/3 + √3/3 = 4√3/3

Now, adding 3√2/2 and 4√3/3. To add these, convert to common denominator 6:

3√2/2 = 9√2/6

4√3/3 = 8√3/6

Adding gives 9√2 + 8√3 over 6. Yep, that's correct. So, a = 9, b = 8, c = 6. Sum is 23.
</execution_5.2>
</Execution_5>

<Final_answer>
Therefore, the sum of $a$, $b$, and $c$ is $\boxed{23}$.
</Final_answer>
</think>