The question asks to "rationalize the denominators" of the given fractions. To rationalize means to eliminate the square root from the denominator by multiplying both the numerator and denominator by an appropriate term. Let's go through each option.

a) $\frac{1}{\sqrt{3}}$

Multiply both the numerator and denominator by $\sqrt{3}$: $\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}.$ Thus, the rationalized form is: $\frac{\sqrt{3}}{3}.$

b) $\frac{5}{\sqrt{5}}$

Multiply both the numerator and denominator by $\sqrt{5}$: $\frac{5}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{5\sqrt{5}}{5} = \sqrt{5}.$ So, the rationalized form simplifies to: $\sqrt{5}.$

c) $\frac{x^2}{\sqrt{x}}$

Multiply both the numerator and denominator by $\sqrt{x}$: $\frac{x^2}{\sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{x^2 \cdot \sqrt{x}}{x} = x\sqrt{x}.$ The rationalized form is: $x\sqrt{x}.$

d) $\frac{4}{\sqrt{2^3}} = \frac{4}{\sqrt{8}}$

First, simplify $\sqrt{8}$ as $\sqrt{4 \times 2} = 2\sqrt{2}$. Now rationalize by multiplying both the numerator and denominator by $\sqrt{2}$: $\frac{4}{2\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{4} = \sqrt{2}.$ Thus, the rationalized form is: $\sqrt{2}.$

Would you like further details or have any questions?

Here are five related questions:

1. What is the general method for rationalizing a denominator with a square root?
2. How would you rationalize a fraction where the denominator contains a sum of square roots?
3. Can you apply the same technique for cube roots in the denominator?
4. How does rationalizing the denominator affect the value of a fraction?
5. What would happen if both the numerator and the denominator have square roots?

Tip: Rationalizing helps simplify expressions for easier computation, especially when dealing with irrational denominators.