Okay, let's see. The problem is to express the sum √2 + 1/√2 + √3 + 1/√3 in the form (a√2 + b√3)/c, where a, b, c are positive integers with c as small as possible. Then find a + b + c. Hmm. Alright, let me break this down step by step.

First, I need to combine all these terms into a single fraction. Let me write down each term:

√2, 1/√2, √3, 1/√3.

I need to combine these into the form (a√2 + b√3)/c. So, maybe I can combine the terms with √2 and the terms with √3 separately first, and then combine those two results into a single fraction.

Let's start with the √2 terms: √2 and 1/√2. Let me add those together. But to add them, they need a common denominator. The denominators here are 1 and √2. So, maybe I can write √2 as √2/1 and 1/√2 as 1/√2. To add these, I need a common denominator. The common denominator would be √2. So, multiplying the first term by √2/√2, we get (√2 * √2)/√2 = (2)/√2. Then adding 1/√2 gives (2 + 1)/√2 = 3/√2.

Wait, hold on, that seems off. Let me check again. If I have √2 + 1/√2, that's equal to √2 + 1/√2. To combine them, yes, common denominator is √2. So:

√2 = (√2 * √2)/√2 = 2/√2. Then 1/√2 is just 1/√2. So adding them gives (2 + 1)/√2 = 3/√2. Hmm, okay, that works. So the √2 terms add up to 3/√2.

Similarly, the √3 terms are √3 and 1/√3. Let's do the same. √3 can be written as √3/1, and 1/√3 is already over √3. So common denominator here is √3. Multiply the first term by √3/√3: (√3 * √3)/√3 = 3/√3. Then add 1/√3: (3 + 1)/√3 = 4/√3.

So now, we have the two combined fractions: 3/√2 and 4/√3. Now, we need to add these two fractions together. To add 3/√2 and 4/√3, we need a common denominator. The denominators are √2 and √3. The least common denominator (LCD) would be √2 * √3 = √6. So, we need to express both fractions with denominator √6.

For 3/√2: Multiply numerator and denominator by √3 to get (3√3)/(√2 * √3) = 3√3/√6.

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

Now, adding them together: 3√3/√6 + 4√2/√6 = (3√3 + 4√2)/√6.

But the problem asks for the expression in the form (a√2 + b√3)/c. However, the denominator here is √6, which is not rational. So, to rationalize the denominator, we need to multiply numerator and denominator by √6.

Let's do that:

Numerator: (3√3 + 4√2) * √6 = 3√3 * √6 + 4√2 * √6.

Simplify each term:

3√3 * √6 = 3√(3*6) = 3√18 = 3*3√2 = 9√2.

4√2 * √6 = 4√(2*6) = 4√12 = 4*2√3 = 8√3.

So numerator becomes 9√2 + 8√3.

Denominator: √6 * √6 = 6.

So, putting it together: (9√2 + 8√3)/6.

Therefore, the expression is (9√2 + 8√3)/6. Now, check if this is in the simplest form with c as small as possible. The denominator is 6, which is already an integer, and c is 6. Let's verify if we can reduce the fractions further.

Looking at the numerator terms: 9√2 and 8√3. The coefficients 9 and 8, and the denominator 6. The greatest common divisor (GCD) of 9, 8, and 6 is 1. Because 9 and 8 are coprime, and 6 shares a factor of 1 with both. So, we can't simplify the fractions any further. Therefore, the expression is in its simplest form with c = 6, a = 9, b = 8.

Therefore, a + b + c = 9 + 8 + 6 = 23.

Wait, but let me double-check my steps to make sure I didn't make a mistake. Let me recap:

1. Combined √2 + 1/√2 = 3/√2.
2. Combined √3 + 1/√3 = 4/√3.
3. Then added 3/√2 and 4/√3 by converting to common denominator √6, resulting in (3√3 + 4√2)/√6.
4. Rationalized the denominator by multiplying numerator and denominator by √6, resulting in (9√2 + 8√3)/6.
5. Checked that GCD of 9, 8, 6 is 1, so c can't be reduced further.

Yes, that seems right. Let me check the arithmetic again:

3/√2 + 4/√3:

Convert to common denominator √6:

3/√2 = 3√3 / √6

4/√3 = 4√2 / √6

Adding them gives (3√3 + 4√2)/√6. Then multiply numerator and denominator by √6:

Numerator: 3√3 * √6 = 3√18 = 3*3√2 = 9√2

4√2 * √6 = 4√12 = 4*2√3 = 8√3

Denominator: √6 * √6 = 6

Yes, correct. So the final expression is (9√2 + 8√3)/6. Therefore, a = 9, b = 8, c = 6. Sum is 23.

I think that's solid. Let me just verify with another approach. Let's compute the original sum numerically and then compute the final expression to see if they match.

Original sum:

√2 ≈ 1.4142

1/√2 ≈ 0.7071

√3 ≈ 1.7320

1/√3 ≈ 0.5774

Adding them up: 1.4142 + 0.7071 = 2.1213; 1.7320 + 0.5774 = 2.3094; total sum ≈ 2.1213 + 2.3094 ≈ 4.4307.

Final expression: (9√2 + 8√3)/6.

Compute numerator:

9√2 ≈ 9 * 1.4142 ≈ 12.7278

8√3 ≈ 8 * 1.7320 ≈ 13.8560

Total numerator ≈ 12.7278 + 13.8560 ≈ 26.5838

Divide by 6: 26.5838 / 6 ≈ 4.4306, which matches the original sum. So the answer is correct.
Therefore, the final answer is 23.
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