Determine the product of $\left(46.2 \times 10^{-1}\right) \cdot\left(5.7 \times 10^{-6}\right)$. Write your answer in scientific notation.
A) $2.6334 \times 10^{-5}$
B) $2633.4 \times 10^{-5}$
C) $2.6334 \times 10^{-1}$
D) $2.6334 \times 10^{-7}$
Answer (1)
Multiply the coefficients: 46.2 × 5.7 = 263.34 .
Multiply the powers of 10: 1 0 − 1 × 1 0 − 6 = 1 0 − 7 .
Combine the results: 263.34 × 1 0 − 7 .
Convert to scientific notation: 2.6334 × 1 0 − 5 .
The final answer is 2.6334 × 1 0 − 5 .
Explanation
Understanding the Problem We are given the expression ( 46.2 × 1 0 − 1 ) ⋅ ( 5.7 × 1 0 − 6 ) and asked to find the product and express it in scientific notation.
Multiplying Coefficients First, we multiply the coefficients: 46.2 × 5.7 = 263.34 .
Multiplying Powers of 10 Next, we multiply the powers of 10: 1 0 − 1 × 1 0 − 6 = 1 0 − 1 + ( − 6 ) = 1 0 − 7 .
Combining Results So, the product is 263.34 × 1 0 − 7 . To write this in scientific notation, we need to express the coefficient as a number between 1 and 10. We can write 263.34 as 2.6334 × 1 0 2 .
Final Answer in Scientific Notation Therefore, the product in scientific notation is ( 2.6334 × 1 0 2 ) × 1 0 − 7 = 2.6334 × 1 0 2 + ( − 7 ) = 2.6334 × 1 0 − 5 .
Conclusion The product of ( 46.2 × 1 0 − 1 ) ⋅ ( 5.7 × 1 0 − 6 ) in scientific notation is 2.6334 × 1 0 − 5 .
Examples
Scientific notation is extremely useful in fields like astronomy and chemistry, where dealing with very large or very small numbers is common. For example, the distance to a star might be 4.5 × 1 0 16 meters, and the size of an atom might be 1.2 × 1 0 − 10 meters. Multiplying these numbers is much easier when they are in scientific notation. In computer science, scientific notation can help represent floating-point numbers, allowing computers to handle a wide range of values efficiently. This notation simplifies calculations and makes it easier to compare vastly different quantities.
