1. The problem is to multiply two numbers in scientific notation: $$(4.001 \times 10^{4}) \times (3.3 \times 10^{3})$$. 2. Recall the rule for multiplying numbers in scientific notation: multiply the decimal parts and add the exponents of 10. 3. Multiply the decimal parts: $$4.001 \times 3.3 = 13.2033$$ 4. Add the exponents of 10: $$10^{4} \times 10^{3} = 10^{4+3} = 10^{7}$$ 5. Combine the results: $$13.2033 \times 10^{7}$$ 6. Since the decimal part must be between 1 and 10 in scientific notation, convert 13.2033 to $1.32033 \times 10^{1}$: $$13.2033 = 1.32033 \times 10^{1}$$ 7. Substitute back: $$1.32033 \times 10^{1} \times 10^{7} = 1.32033 \times 10^{8}$$ 8. Final answer in scientific notation: $$\boxed{1.32033 \times 10^{8}}$$