Rationalize $\frac{\sqrt{6} - \sqrt{4}}{\sqrt{6} + \sqrt{4}}$

Analysis

To rationalize a fraction with a radical expression in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $(\sqrt{6} + \sqrt{4})$ is $(\sqrt{6} - \sqrt{4})$.

Concept

Rationalization is used to remove square roots from the denominator. Recall the algebraic identity: $(a + b)(a - b) = a^2 - b^2$

Step-by-Step Solution

Step 1: Set up the multiplication

Step 2: Expand the numerator The numerator is $(\sqrt{6} - \sqrt{4})(\sqrt{6} - \sqrt{4}) = (\sqrt{6} - \sqrt{4})^2$. Using the expansion $(a - b)^2 = a^2 - 2ab + b^2$: $= (\sqrt{6})^2 - 2(\sqrt{6})(\sqrt{4}) + (\sqrt{4})^2$ $= 6 - 2\sqrt{24} + 4$ $= 10 - 2\sqrt{4 \times 6}$ $= 10 - 2(2\sqrt{6})$ $= 10 - 4\sqrt{6}$

Step 3: Expand the denominator Using the difference of two squares $(a + b)(a - b) = a^2 - b^2$: $= (\sqrt{6})^2 - (\sqrt{4})^2$ $= 6 - 4$ $= 2$

Step 4: Simplify the fraction

Divide both terms in the numerator by 2: $= \frac{10}{2} - \frac{4\sqrt{6}}{2}$ $= 5 - 2\sqrt{6}$

Conclusion

Comparing our result to the given options, we find that it matches option D.

Exam Tip

When rationalizing, always check if the square roots can be simplified further (like $\sqrt{4} = 2$ or $\sqrt{24} = 2\sqrt{6}$) before or during the calculation to keep your numbers small and manageable. This prevents errors during multiplication!