Simplify $\frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}$

To simplify the expression $\frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}$, we use the method of rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator.

1. Identify the Conjugate

The denominator is $\sqrt{2} + \sqrt{3}$. Its conjugate is $\sqrt{2} - \sqrt{3}$.

2. Set up the Multiplication

Multiply the numerator and denominator by $(\sqrt{2} - \sqrt{3})$:

3. Expand the Numerator

Using the FOIL method (First, Outer, Inner, Last): $(2\sqrt{2} - \sqrt{3})(\sqrt{2} - \sqrt{3})$ $= (2\sqrt{2} \times \sqrt{2}) - (2\sqrt{2} \times \sqrt{3}) - (\sqrt{3} \times \sqrt{2}) + (\sqrt{3} \times \sqrt{3})$ $= (2 \times 2) - 2\sqrt{6} - \sqrt{6} + 3$ $= 4 - 3\sqrt{6} + 3$ $= 7 - 3\sqrt{6}$

4. Expand the Denominator

Using the difference of two squares formula $(a+b)(a-b) = a^2 - b^2$: $(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3})$ $= (\sqrt{2})^2 - (\sqrt{3})^2$ $= 2 - 3$ $= -1$

5. Combine and Simplify

Now, place the expanded numerator over the expanded denominator:

Divide each term by $-1$: $= \frac{7}{-1} - \frac{3\sqrt{6}}{-1}$ $= -7 + 3\sqrt{6}$ $= 3\sqrt{6} - 7$

Comparing this to the given options, it matches option A.

Exam Tip: When rationalizing, always remember that the denominator will result in a simple integer (or a negative integer). If you end up with a square root in your denominator, you haven't finished the process! Also, be very careful with your signs when subtracting the terms in the numerator.