1. The problem asks to simplify the expression $$\sqrt{x^2 + y^2}$$ and find which option it equals. 2. Recall that $$\sqrt{a^2} = |a|$$ for any real number $$a$$ 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 $$\sqrt{a^2 + b^2} \neq |a| + |b|$$ in general. 4. Let's analyze the options: - a. $$x + y$$ is incorrect because $$\sqrt{x^2 + y^2}$$ is not a simple sum. - b. $$|x| + |y|$$ is incorrect because the square root of sum of squares is not the sum of absolute values. - c. $$x^2 y^2 \sqrt{\frac{1}{x^2} + \frac{1}{y^2}}$$ is complicated and does not simplify to the original expression. - d. $$|x| \sqrt{1 + \frac{y^2}{x^2}}$$ can be rewritten as $$|x| \sqrt{\frac{x^2 + y^2}{x^2}} = |x| \frac{\sqrt{x^2 + y^2}}{|x|} = \sqrt{x^2 + y^2}$$ which matches the original expression. 5. Therefore, the correct answer is option d. Final answer: $$\boxed{d}$$