1. **State the problem:** Simplify the expression $$\sqrt{\frac{75}{30}}$$. 2. **Recall the property of square roots:** $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ where $a$ and $b$ are positive numbers. 3. **Apply the property:** $$\sqrt{\frac{75}{30}} = \frac{\sqrt{75}}{\sqrt{30}}$$ 4. **Simplify the square roots by factoring:** - $75 = 25 \times 3$ - $30 = 15 \times 2$ So, $$\frac{\sqrt{25 \times 3}}{\sqrt{15 \times 2}} = \frac{\sqrt{25} \times \sqrt{3}}{\sqrt{15} \times \sqrt{2}}$$ 5. **Calculate the square roots of perfect squares:** $$\frac{5 \sqrt{3}}{\sqrt{15} \times \sqrt{2}}$$ 6. **Combine the denominator square roots:** $$\frac{5 \sqrt{3}}{\sqrt{15 \times 2}} = \frac{5 \sqrt{3}}{\sqrt{30}}$$ 7. **Rationalize the denominator:** Multiply numerator and denominator by $\sqrt{30}$: $$\frac{5 \sqrt{3} \times \sqrt{30}}{\sqrt{30} \times \sqrt{30}} = \frac{5 \sqrt{90}}{30}$$ 8. **Simplify $\sqrt{90}$:** $$\sqrt{90} = \sqrt{9 \times 10} = 3 \sqrt{10}$$ So, $$\frac{5 \times 3 \sqrt{10}}{30} = \frac{15 \sqrt{10}}{30}$$ 9. **Simplify the fraction:** $$\frac{\cancel{15} \sqrt{10}}{\cancel{30}} = \frac{1 \sqrt{10}}{2} = \frac{\sqrt{10}}{2}$$ **Final answer:** $$\boxed{\frac{\sqrt{10}}{2}}$$