1. The problem asks to simplify the expression $$\sqrt{x^2 + y^2}$$ and find which option it equals. 2. Recall that $$\sqrt{x^2} = |x|$$ because the square root function returns the non-negative value. 3. The expression $$\sqrt{x^2 + y^2}$$ cannot be simplified to $$x + y$$ or $$|x| + |y|$$ because $$\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$$ in general. 4. Let's analyze option d: $$|x| \sqrt{1 + \frac{y^2}{x^2}} = |x| \sqrt{\frac{x^2 + y^2}{x^2}} = |x| \frac{\sqrt{x^2 + y^2}}{|x|} = \sqrt{x^2 + y^2}$$. 5. Therefore, option d is equivalent to the original expression. 6. The other options do not simplify correctly to $$\sqrt{x^2 + y^2}$$. Final answer: d. $$|x| \sqrt{1 + \frac{y^2}{x^2}}$$