Problem 93 - College Algebra Essentials

Question

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. $$\left(4.3 \times 10^{8}\right)\left(6.2 \times 10^{4}\right)$$

Step-by-Step Solution

Answer

The result of \((4.3 \times 10^{8}) \times (6.2 \times 10^{4})\) is \(2.67 \times 10^{13}\).

Step 1: Multiplication of Numeric Factors

Multiply the numeric parts of the scientific notations: \(4.3\) and \(6.2\), which gives \(26.66\).

Step 2: Multiplication of Power-of-10 Factors

Multiply the power-of-10 parts by adding the exponents: \(10^{8}\) and \(10^{4}\) using the rule \(10^{m} \cdot 10^{n} = 10^{m+n}\), hence \(10^{8+4} = 10^{12}\).

Step 3: Write the Result in Scientific Notation

Combine the results from steps 1 and 2 to express the final result in scientific notation: \(26.66 \times 10^{12}\).

Step 4: Convert the Answer to Correct Scientific Notation

The number \(26.66 \times 10^{12}\) is not in proper scientific notation because the digit term is not between 1 (inclusive) and 10 (exclusive). To correct this, convert it into \(2.666 \times 10^{13}\).

Step 5: Final Answer

Finally, round the decimal factor to two decimal places to get \(2.67 \times 10^{13}\)