27⅖ - 5⅓ + 4⅔

We can rewrite 27 as $3^3$.

$$27^{1/6} = (3^3)^{1/6} = 3^{3/6} = 3^{1/2} = \sqrt{3}$$

This can be written as the cube root of $5^2$.

$$5^{2/3} = \sqrt[3]{5^2} = \sqrt[3]{25}$$

This term cannot be simplified further as 25 is not a perfect cube.

We can rewrite 4 as $2^2$.

$$4^{3/4} = (2^2)^{3/4} = 2^{2 \times \frac{3}{4}} = 2^{\frac{6}{4}} = 2^{\frac{3}{2}}$$

Now, convert this back to radical form:

$$2^{3/2} = \sqrt{2^3} = \sqrt{8}$$

Simplify $\sqrt{8}$:

$$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$$

Substitute the simplified values back into the original expression:

$$27^{1/6} - 5^{2/3} + 4^{3/4} = \sqrt{3} - \sqrt[3]{25} + 2\sqrt{2}$$

These are unlike radicals (different indices and radicands), so they cannot be combined further.

The final answer is $\sqrt{3} - \sqrt[3]{25} + 2\sqrt{2}$.