The problem in the image asks for the rationalization of the expression:

$\frac{14 \sqrt{2}}{\sqrt{5} - 2\sqrt{3}}$

Steps to Rationalize:

1. Multiply by the Conjugate: To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of $\sqrt{5} - 2\sqrt{3}$ is $\sqrt{5} + 2\sqrt{3}$.

   So, we multiply as follows:

   $\frac{14 \sqrt{2}}{\sqrt{5} - 2\sqrt{3}} \times \frac{\sqrt{5} + 2\sqrt{3}}{\sqrt{5} + 2\sqrt{3}} = \frac{14 \sqrt{2} (\sqrt{5} + 2 \sqrt{3})}{(\sqrt{5})^2 - (2 \sqrt{3})^2}$

2. Simplify the Denominator: The denominator is a difference of squares:

   $(\sqrt{5})^2 - (2 \sqrt{3})^2 = 5 - 4 \times 3 = 5 - 12 = -7$

3. Expand the Numerator: Now, expand the numerator:

   $14 \sqrt{2} (\sqrt{5} + 2 \sqrt{3}) = 14 \sqrt{2} \times \sqrt{5} + 14 \sqrt{2} \times 2 \sqrt{3}$

   Simplifying each term:

   $14 \sqrt{2} \times \sqrt{5} = 14 \sqrt{10}$ $14 \sqrt{2} \times 2 \sqrt{3} = 28 \sqrt{6}$

   Therefore, the numerator becomes:

   $14 \sqrt{10} + 28 \sqrt{6}$

4. Combine and Simplify: Now, we combine everything:

   $\frac{14 \sqrt{10} + 28 \sqrt{6}}{-7} = -2 \sqrt{10} - 4 \sqrt{6}$

Thus, the rationalized form of the given expression is:

$\boxed{-2 \sqrt{10} - 4 \sqrt{6}}$

This matches option a in the problem.

Would you like a deeper explanation or any other questions?

Related Questions:

1. How do you find the conjugate of a binomial with square roots?
2. Why is multiplying by the conjugate useful in rationalizing denominators?
3. How do you simplify expressions involving products of square roots?
4. What is the difference of squares formula, and why does it help with rationalizing?
5. Can you rationalize a more complex denominator, such as one involving cube roots?

Tip:

Always double-check both numerator and denominator after rationalization to ensure that the expression is fully simplified.