$\left[ \frac{3\sqrt{2}}{\sqrt{6} - \sqrt{3}} - \frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}} - \frac{6}{\sqrt{8} - \sqrt{12}} \right] = $ $x$

### Concept & Formula This expression is a test of your ability to accurately apply **rationalization** across multiple terms. The standard procedure is to multiply the numerator and denominator of each fraction by the conjugate of its denominator. Additionally, we use the multiplication property of surds: $$ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} $$ ### Step-by-Step Solution * **Calculation / Deduction:** * We will rationalize each of the three terms individually. * **First Term:** $$ \frac{3\sqrt{2}}{\sqrt{6} - \sqrt{3}} $$ Multiply by the conjugate $(\sqrt{6} + \sqrt{3})$: $$ = \frac{3\sqrt{2}(\sqrt{6} + \sqrt{3})}{(\sqrt{6})^2 - (\sqrt{3})^2} $$ $$ = \frac{3\sqrt{12} + 3\sqrt{6}}{6 - 3} $$ $$ = \frac{3(2\sqrt{3}) + 3\sqrt{6}}{3} = \frac{6\sqrt{3} + 3\sqrt{6}}{3} = 2\sqrt{3} + \sqrt{6} $$ * **Second Term:** $$ \frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}} $$ Multiply by the conjugate $(\sqrt{6} + \sqrt{2})$: $$ = \frac{4\sqrt{3}(\sqrt{6} + \sqrt{2})}{(\sqrt{6})^2 - (\sqrt{2})^2} $$ $$ = \frac{4\sqrt{18} + 4\sqrt{6}}{6 - 2} $$ $$ = \frac{4(3\sqrt{2}) + 4\sqrt{6}}{4} = \frac{12\sqrt{2} + 4\sqrt{6}}{4} = 3\sqrt{2} + \sqrt{6} $$ * **Third Term:** $$ \frac{6}{\sqrt{8} - \sqrt{12}} $$ First, note that the denominator will be negative ($8 - 12 = -4$). Multiply by the conjugate $(\sqrt{8} + \sqrt{12})$: $$ = \frac{6(\sqrt{8} + \sqrt{12})}{(\sqrt{8})^2 - (\sqrt{12})^2} $$ $$ = \frac{6(2\sqrt{2} + 2\sqrt{3})}{8 - 12} $$ $$ = \frac{12\sqrt{2} + 12\sqrt{3}}{-4} = -3\sqrt{2} - 3\sqrt{3} $$ * **Combining All Terms:** Substitute the simplified terms back into the original expression carefully: $$ (2\sqrt{3} + \sqrt{6}) - (3\sqrt{2} + \sqrt{6}) - (-3\sqrt{2} - 3\sqrt{3}) $$ * Distribute the negative signs: $$ = 2\sqrt{3} + \sqrt{6} - 3\sqrt{2} - \sqrt{6} + 3\sqrt{2} + 3\sqrt{3} $$ * Group like terms together: $$ = (2\sqrt{3} + 3\sqrt{3}) + (\sqrt{6} - \sqrt{6}) + (-3\sqrt{2} + 3\sqrt{2}) $$ $$ = 5\sqrt{3} + 0 + 0 = 5\sqrt{3} $$ ### Exam Strategy & Shortcut Always look to simplify surds *before* distributing them, if possible. For instance, in the third term $\sqrt{8} - \sqrt{12}$, extracting the common factor transforms it to $2(\sqrt{2} - \sqrt{3})$. This makes the fraction $\frac{6}{2(\sqrt{2} - \sqrt{3})} = \frac{3}{\sqrt{2} - \sqrt{3}}$, which rationalizes cleanly to $3(\sqrt{2} + \sqrt{3})$ over $-1$. Finding these smaller simplifications reduces the multiplication numbers and speeds up the calculation. ### Common Pitfall The greatest risk here is the third term, because the denominator $8 - 12$ results in a negative integer ($-4$). When this negative integer is divided into the positive numerator, it changes the signs of the result to negative ($-3\sqrt{2} - 3\sqrt{3}$). Furthermore, this negative result is being *subtracted* in the main equation, flipping the signs back to positive. Keeping track of these double negatives is crucial. ### Final Answer **Therefore, the correct answer is $5\sqrt{3}$.**