The question appears to involve rationalizing the denominator of the following expressions:

1. $\frac{2\sqrt{10}}{\sqrt{7} - \sqrt{5}}$
2. $\frac{\sqrt{5}}{\sqrt{8}}$

Let's solve both parts by rationalizing the denominators.

Part (a): $\frac{2\sqrt{10}}{\sqrt{7} - \sqrt{5}}$

To rationalize this, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{7} - \sqrt{5}$ is $\sqrt{7} + \sqrt{5}$.

$\frac{2\sqrt{10}}{\sqrt{7} - \sqrt{5}} \times \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} + \sqrt{5}} = \frac{2\sqrt{10}(\sqrt{7} + \sqrt{5})}{(\sqrt{7})^2 - (\sqrt{5})^2}$

Simplify the denominator: $(\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2$

Now, the expression becomes: $\frac{2\sqrt{10}(\sqrt{7} + \sqrt{5})}{2}$

Cancel out the 2 in the numerator and denominator: $\sqrt{10}(\sqrt{7} + \sqrt{5}) = \sqrt{70} + \sqrt{50}$

So, the final answer is: $\boxed{\sqrt{70} + \sqrt{50}}$

Since $\sqrt{50} = 5\sqrt{2}$, we can simplify further: $\boxed{\sqrt{70} + 5\sqrt{2}}$

Part (b): $\frac{\sqrt{5}}{\sqrt{8}}$

We can rationalize the denominator by simplifying the square root of 8: $\sqrt{8} = 2\sqrt{2}$

So, the expression becomes: $\frac{\sqrt{5}}{2\sqrt{2}}$

Now, multiply both the numerator and the denominator by $\sqrt{2}$ to rationalize the denominator: $\frac{\sqrt{5}}{2\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{10}}{2 \times 2} = \frac{\sqrt{10}}{4}$

So, the final answer is: $\boxed{\frac{\sqrt{10}}{4}}$

Would you like more details on any of these steps?

Related questions:

1. How do you multiply radicals when rationalizing the denominator?
2. What are conjugates, and how are they used in rationalizing expressions?
3. Can you apply the same method for rationalizing complex numbers?
4. Why is it important to rationalize denominators?
5. How do you simplify radicals like $\sqrt{50}$ or $\sqrt{8}$?

Tip: When rationalizing denominators, always look for the conjugate of the denominator in case of binomial radicals.