Determine the product. Write your answer in scientific notation. $(15.4 \times 10^2) \cdot(2.8 \times 10^{-4})=$ ?
Answer (1)
Multiply the decimal parts: 15.4 × 2.8 = 43.12 .
Multiply the powers of 10: 1 0 2 × 1 0 − 4 = 1 0 − 2 .
Combine the results: 43.12 × 1 0 − 2 .
Convert to scientific notation: 4.312 × 1 0 − 1 .
The final answer is 4.312 × 1 0 − 1 .
Explanation
Understanding the Problem We are given the expression ( 15.4 × 1 0 2 ) ⋅ ( 2.8 × 1 0 − 4 ) and asked to determine the product and write the answer in scientific notation.
Multiplying Decimal Parts First, let's multiply the decimal parts: 15.4 × 2.8 = 43.12 .
Multiplying Powers of 10 Next, let's multiply the powers of 10: 1 0 2 × 1 0 − 4 = 1 0 2 + ( − 4 ) = 1 0 − 2 .
Combining Results Now, combine the results: 43.12 × 1 0 − 2 .
Converting to Scientific Notation To express this in scientific notation, we need the decimal part to be between 1 and 10. So, we rewrite 43.12 as 4.312 × 1 0 1 .
Final Calculation Therefore, the expression becomes ( 4.312 × 1 0 1 ) × 1 0 − 2 = 4.312 × 1 0 1 + ( − 2 ) = 4.312 × 1 0 − 1 .
Final Answer The product in scientific notation is 4.312 × 1 0 − 1 .
Examples
Scientific notation is used in many fields, such as physics and astronomy, to represent very large or very small numbers. For example, the speed of light is approximately 3.0 × 1 0 8 meters per second, and the size of an atom is approximately 1.0 × 1 0 − 10 meters. Using scientific notation makes it easier to work with these numbers and perform calculations.
