1. **State the problem:** Rationalize the denominator and simplify the expression $$\sqrt{\frac{5}{x^3}}$$. 2. **Rewrite the expression:** We can write the square root of a fraction as the fraction of square roots: $$\sqrt{\frac{5}{x^3}} = \frac{\sqrt{5}}{\sqrt{x^3}}$$. 3. **Simplify the denominator:** Since $$\sqrt{x^3} = \sqrt{x^2 \cdot x} = \sqrt{x^2} \cdot \sqrt{x} = x \sqrt{x}$$ (assuming $$x > 0$$ for real values). 4. **Rewrite the expression:** $$\frac{\sqrt{5}}{x \sqrt{x}}$$. 5. **Rationalize the denominator:** To eliminate the square root in the denominator, multiply numerator and denominator by $$\sqrt{x}$$: $$\frac{\sqrt{5}}{x \sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{\sqrt{5} \cdot \sqrt{x}}{x \sqrt{x} \cdot \sqrt{x}} = \frac{\sqrt{5x}}{x \cdot x} = \frac{\sqrt{5x}}{x^2}$$. 6. **Final simplified expression:** $$\boxed{\frac{\sqrt{5x}}{x^2}}$$. This is the rationalized and fully simplified form of the original expression.