Okay, let's tackle this problem step by step. So, the function f(n) is defined in two cases: if n is greater than or equal to 4, then f(n) is the floor of n; if n is less than 4, it's the ceiling of n. We need to compute f(π/3) + f(√45) + f(8^(2/3)). Hmm, let's break this down term by term.

First, let's find each value inside the function one by one. Starting with π/3. I know π is approximately 3.1416, so dividing that by 3 gives roughly 1.0472. Since 1.0472 is definitely less than 4, we use the ceiling function here. The ceiling of 1.0472 should be 2, right? Wait, ceiling rounds up to the nearest integer. Since 1.0472 is between 1 and 2, the ceiling would be 2. So f(π/3) = 2. That seems straightforward.

Next term is √45. Let me calculate that. The square of 6 is 36, and 7 squared is 49, so √45 should be between 6 and 7. Let's see, 6.7 squared is 44.89, which is close to 45. So √45 is approximately 6.7082. Since 6.7082 is greater than 4, we use the floor function here. The floor of 6.7082 is 6. So f(√45) = 6. That makes sense.

Third term is 8^(2/3). Let's compute that. 8 is 2^3, so 8^(1/3) is 2. Then raising that to the power of 2 gives 2^2 = 4. Wait, 8^(2/3) is the same as (8^(1/3))^2, which is 2^2 = 4. So 8^(2/3) equals exactly 4. Since 4 is equal to 4, we use the floor function here. The floor of 4 is 4. So f(8^(2/3)) = 4. That's clear.

Now, adding them all up: 2 + 6 + 4. Let's do that step by step. 2 + 6 is 8, and 8 + 4 is 12. So the total sum is 12. Hmm, that seems straightforward. Let me double-check each term to make sure I didn't make a mistake.

First term: π/3 ≈ 1.0472, ceiling is 2. Correct. Second term: √45 ≈ 6.7082, floor is 6. Correct. Third term: 8^(2/3) is exactly 4, floor is 4. Correct. Sum is 2 + 6 + 4 = 12. Yep, that seems right.

Wait, just to be thorough, let me verify each calculation again. For π/3, π is approximately 3.1416, divided by 3 is roughly 1.0472. Since 1.0472 is less than 4, ceiling is indeed 2. No issues there.

For √45: 6.7082 is correct? Let's compute 6.7^2 = 44.89, 6.7082^2 is 6.7082 * 6.7082. Let's compute 6 * 6 = 36, 6 * 0.7082 = 4.2492, 0.7082 * 6 = 4.2492, 0.7082 * 0.7082 ≈ 0.5015. Adding those up: 36 + 4.2492 + 4.2492 + 0.5015 ≈ 45. So yes, √45 is approximately 6.7082, which is correct. Since it's greater than 4, floor is 6.

Third term: 8^(2/3). Let's compute 8^(1/3) first. 8 is 2^3, so cube root of 8 is 2. Then 2 squared is 4. So 8^(2/3) is exactly 4. Since 4 is equal to 4, floor is 4. Correct.

Sum: 2 + 6 + 4. 2 + 6 is 8, 8 + 4 is 12. All right, that seems solid. I can't see any mistakes here. So the final answer should be 12.

**Final Answer**
\boxed{12}
Therefore, the sum is $\boxed{12}$.
</think>