1. **State the problem:** Evaluate the expression $$\frac{\sqrt{900 - \left(\frac{30}{14}\right)^2}}{13}$$. 2. **Recall the formula and rules:** The square root function is defined as $$\sqrt{x}$$ where $$x \geq 0$$. We must first simplify inside the square root before dividing by 13. 3. **Calculate the square inside the root:** $$\left(\frac{30}{14}\right)^2 = \frac{30^2}{14^2} = \frac{900}{196}$$. 4. **Subtract inside the root:** $$900 - \frac{900}{196} = \frac{900 \times 196}{196} - \frac{900}{196} = \frac{176400 - 900}{196} = \frac{175500}{196}$$. 5. **Simplify the fraction inside the root:** $$\frac{175500}{196}$$. 6. **Take the square root:** $$\sqrt{\frac{175500}{196}} = \frac{\sqrt{175500}}{\sqrt{196}} = \frac{\sqrt{175500}}{14}$$. 7. **Simplify $$\sqrt{175500}$$:** Factor 175500: $$175500 = 1755 \times 100 = 3 \times 5 \times 7 \times 5 \times 7 \times 100$$. More precisely, $$175500 = 3 \times 5^2 \times 7^2 \times 100$$. So, $$\sqrt{175500} = \sqrt{3 \times 5^2 \times 7^2 \times 100} = 5 \times 7 \times 10 \times \sqrt{3} = 350 \sqrt{3}$$. 8. **Substitute back:** $$\frac{\sqrt{175500}}{14} = \frac{350 \sqrt{3}}{14} = 25 \sqrt{3}$$. 9. **Divide by 13:** $$\frac{25 \sqrt{3}}{13}$$. **Final answer:** $$\boxed{\frac{25 \sqrt{3}}{13}}$$