1. **State the problem:** Calculate the value of $$r = 2 \times 0.3 \times 0.09 \times \sqrt{\frac{0.002 \times 0.09}{9 \times 10^{9}}}$$. 2. **Recall the formula and rules:** The expression involves multiplication and a square root. Remember that multiplication is associative and commutative, and the square root of a fraction is the square root of numerator divided by the square root of denominator. 3. **Calculate inside the square root first:** $$0.002 \times 0.09 = 0.00018$$ $$9 \times 10^{9} = 9 \times 10^{9}$$ So, $$\frac{0.00018}{9 \times 10^{9}} = 2 \times 10^{-14}$$ 4. **Calculate the square root:** $$\sqrt{2 \times 10^{-14}} = \sqrt{2} \times \sqrt{10^{-14}} = 1.4142 \times 10^{-7}$$ 5. **Multiply all factors:** $$2 \times 0.3 = 0.6$$ $$0.6 \times 0.09 = 0.054$$ $$0.054 \times 1.4142 \times 10^{-7} = 7.63668 \times 10^{-9}$$ 6. **Final answer:** $$r \approx 7.64 \times 10^{-9}$$