Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. $(4.3\times {10}^8)(6.2\times {10}^4)$

Verified step by step guidance

1. Identify the problem as multiplying two numbers expressed in scientific notation: \((4.3 \times 10^{8})\) and \((6.2 \times 10^{4})\).
2. Recall the rule for multiplying numbers in scientific notation: multiply the decimal factors and add the exponents of 10. So, calculate \((4.3 \times 6.2)\) and add the exponents \$8 + 4$.
3. Perform the multiplication of the decimal factors: \$4.3 \times 6.2$ (do not calculate the exact value here, just set up the expression).
4. Add the exponents of 10: \$8 + 4 = 12\(, so the power of ten part becomes \)10^{12}$.
5. Combine the results to write the product in scientific notation as \((\text{decimal factor}) \times 10^{12}\). If the decimal factor is not between 1 and 10, adjust it accordingly and modify the exponent to keep the value equivalent. Finally, round the decimal factor to two decimal places if necessary.

Key Concepts

Scientific Notation

Scientific notation expresses numbers as a product of a decimal factor and a power of ten, typically in the form a × 10^n, where 1 ≤ a < 10 and n is an integer. It simplifies working with very large or very small numbers, making calculations and comparisons easier.

Multiplication of Numbers in Scientific Notation

To multiply numbers in scientific notation, multiply their decimal factors and add their exponents. For example, (a × 10^m)(b × 10^n) = (a × b) × 10^(m+n). After multiplication, adjust the decimal factor to ensure it remains between 1 and 10.

Rounding Decimal Factors

Rounding decimal factors involves limiting the number of decimal places to a specified precision, here two decimal places. This ensures the answer is concise and meets the problem's requirements, while maintaining reasonable accuracy in scientific notation.