1. **State the problem:** Express the sum $$\frac{1}{x^3} + \frac{1}{2xy} + \frac{1}{x^2 y^2}$$ as a single fraction. 2. **Find the common denominator:** The denominators are $$x^3$$, $$2xy$$, and $$x^2 y^2$$. The least common denominator (LCD) must include the highest powers of each variable and constants: - For $$x$$: highest power is $$x^3$$ - For $$y$$: highest power is $$y^2$$ - For constants: include 2 from $$2xy$$ So, $$\text{LCD} = 2 x^3 y^2$$. 3. **Rewrite each fraction with the LCD:** - $$\frac{1}{x^3} = \frac{1 \cdot 2 y^2}{x^3 \cdot 2 y^2} = \frac{2 y^2}{2 x^3 y^2}$$ - $$\frac{1}{2 x y} = \frac{1 \cdot x^2 y}{2 x y \cdot x^2 y} = \frac{x^2 y}{2 x^3 y^2}$$ - $$\frac{1}{x^2 y^2} = \frac{1 \cdot 2 x}{x^2 y^2 \cdot 2 x} = \frac{2 x}{2 x^3 y^2}$$ 4. **Add the numerators over the common denominator:** $$\frac{2 y^2 + x^2 y + 2 x}{2 x^3 y^2}$$ 5. **Final answer:** The sum expressed as a single fraction is $$\boxed{\frac{2 y^2 + x^2 y + 2 x}{2 x^3 y^2}}$$ This matches option (a).