1. **State the problem:** Calculate the value of the expression $4640 \times 0.083 + 0.0000020 \times 5585000 \sqrt{0.083}$.\n\n2. **Recall formulas and rules:**\n- Multiplication and addition are basic arithmetic operations.\n- The square root function $\sqrt{x}$ gives the positive number whose square is $x$.\n- Order of operations: calculate square root first, then multiplication, then addition.\n\n3. **Calculate the square root:**\n$$\sqrt{0.083} \approx 0.2880$$\n\n4. **Calculate each term:**\n- First term: $4640 \times 0.083 = 385.12$\n- Second term: $0.0000020 \times 5585000 \times 0.2880$\nCalculate $0.0000020 \times 5585000 = 11.17$\nThen multiply by $0.2880$: $11.17 \times 0.2880 = 3.215$\n\n5. **Add the two terms:**\n$$385.12 + 3.215 = 388.335$$\n\n**Final answer:** $388.335$