1. **State the problem:** Multiply the numbers in scientific notation: $$(8.36 \times 10^3)(7.5 \times 10^{-1})$$ 2. **Recall the multiplication rule for scientific notation:** When multiplying numbers in scientific notation, multiply the decimal parts and add the exponents of 10: $$ (a \times 10^m)(b \times 10^n) = (a \times b) \times 10^{m+n} $$ 3. **Multiply the decimal parts:** $$ 8.36 \times 7.5 = 62.7 $$ 4. **Add the exponents:** $$ 3 + (-1) = 2 $$ 5. **Combine the results:** $$ 62.7 \times 10^2 $$ 6. **Convert to proper scientific notation:** Since 62.7 is greater than 10, rewrite as: $$ 62.7 = 6.27 \times 10^1 $$ So, $$ 62.7 \times 10^2 = (6.27 \times 10^1) \times 10^2 = 6.27 \times 10^{1+2} = 6.27 \times 10^3 $$ **Final answer:** $$ \boxed{6.27 \times 10^3} $$