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

Analysis

To rationalize a fraction with a surd in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $(\sqrt{2} - \sqrt{3})$ is $(\sqrt{2} + \sqrt{3})$.

Concept

Rationalization removes the square root from the denominator. We use the difference of two squares identity: $(a - b)(a + b) = a^2 - b^2$.

Step-by-Step Solution

Step 1: Set up the multiplication

Step 2: Expand the numerator Using the FOIL method (First, Outer, Inner, Last): $(\sqrt{2} + \sqrt{3})(\sqrt{2} + \sqrt{3}) = (\sqrt{2})^2 + \sqrt{2}\sqrt{3} + \sqrt{3}\sqrt{2} + (\sqrt{3})^2$ $= 2 + \sqrt{6} + \sqrt{6} + 3$ $= 5 + 2\sqrt{6}$

Step 3: Expand the denominator Using the difference of two squares: $(\sqrt{2} - \sqrt{3})(\sqrt{2} + \sqrt{3}) = (\sqrt{2})^2 - (\sqrt{3})^2$ $= 2 - 3$ $= -1$

Step 4: Combine and simplify

Divide each term by $-1$: $= -5 - 2\sqrt{6}$

Conclusion

Comparing our result to the given options, we find that the calculation matches option A.

Exam Tip

When rationalizing, always check the sign of your denominator after squaring. Many students forget that $(\sqrt{2})^2 - (\sqrt{3})^2$ results in $2 - 3 = -1$, not $1$. If you get a negative denominator, remember to distribute that negative sign to every term in your numerator!