We are given the expression:

$\frac{8}{\sqrt{3}} - \frac{\sqrt{48}}{6} + \frac{4}{\sqrt{27}}$

Step 1: Rationalizing $\frac{8}{\sqrt{3}}$

Multiply numerator and denominator by $\sqrt{3}$:

$\frac{8}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{8\sqrt{3}}{3}$

Step 2: Simplifying $\frac{\sqrt{48}}{6}$

First, simplify $\sqrt{48}$:

$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$

So,

$\frac{\sqrt{48}}{6} = \frac{4\sqrt{3}}{6} = \frac{2\sqrt{3}}{3}$

Step 3: Rationalizing $\frac{4}{\sqrt{27}}$

First, simplify $\sqrt{27}$:

$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$

So,

$\frac{4}{\sqrt{27}} = \frac{4}{3\sqrt{3}}$

Multiply numerator and denominator by $\sqrt{3}$:

$\frac{4\sqrt{3}}{3 \times 3} = \frac{4\sqrt{3}}{9}$

Step 4: Expressing with a Common Denominator

We now have:

$\frac{8\sqrt{3}}{3} - \frac{2\sqrt{3}}{3} + \frac{4\sqrt{3}}{9}$

The first two terms have denominator 3, so:

$\frac{(8 - 2)\sqrt{3}}{3} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$

Now, combine with $\frac{4\sqrt{3}}{9}$, using a common denominator of 9:

$\frac{6\sqrt{3} \times 3}{9} + \frac{4\sqrt{3}}{9} = \frac{18\sqrt{3} + 4\sqrt{3}}{9}$

$= \frac{22\sqrt{3}}{9}$

Final Answer:

$\frac{22\sqrt{3}}{9}$

where $a = 22$, $b = 3$, and $c = 9$.

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Would you like more details on any step?

Related Questions:

1. How do you simplify surds in fraction form?
2. What is the process of rationalizing denominators?
3. How do you find the simplest form of a surd?
4. What are common mistakes when simplifying surds?
5. How do you find a common denominator when adding fractions with surds?

Tip: Always check if the surd in the denominator can be simplified before rationalizing!