Determine the sum. Write your answer in scientific notation.
[tex]37.2 \times 10^4+2.9 \times 10^4[/tex]
Answer (2)
Add the numbers: 37.2 × 1 0 4 + 2.9 × 1 0 4 = ( 37.2 + 2.9 ) × 1 0 4 .
Calculate the sum inside the parentheses: 37.2 + 2.9 = 40.1 .
Express the result: 40.1 × 1 0 4 .
Convert to scientific notation: 4.01 × 1 0 5 . The final answer is 4.01 × 1 0 5 .
Explanation
Problem Analysis We are asked to determine the sum of 37.2 × 1 0 4 and 2.9 × 1 0 4 and express the result in scientific notation.
Calculating the Sum First, we calculate the sum: 37.2 × 1 0 4 + 2.9 × 1 0 4 .
Factoring We can factor out 1 0 4 from both terms: ( 37.2 + 2.9 ) × 1 0 4 .
Addition Now, we add the numbers inside the parentheses: 37.2 + 2.9 = 40.1 .
Result So, the sum is 40.1 × 1 0 4 .
Scientific Notation To express this in scientific notation, we need to write 40.1 as 4.01 × 1 0 1 . Therefore, 40.1 × 1 0 4 = ( 4.01 × 1 0 1 ) × 1 0 4 = 4.01 × ( 1 0 1 × 1 0 4 ) = 4.01 × 1 0 1 + 4 = 4.01 × 1 0 5 .
Final Answer Comparing our result with the given options, we find that the correct answer is 4.01 × 1 0 5 .
Examples
Scientific notation is used in many fields, such as physics and astronomy, to represent very large or very small numbers. For example, the distance to the nearest star, Proxima Centauri, is approximately 4.01 × 1 0 13 kilometers. Similarly, the size of an atom is approximately 1 × 1 0 − 10 meters. Using scientific notation makes it easier to work with these numbers and perform calculations.
To sum 37.2 × 1 0 4 and 2.9 × 1 0 4 , we factor out the common term and add the coefficients to get 40.1 × 1 0 4 . This is then converted to scientific notation as 4.01 × 1 0 5 . Thus, the final answer is 4.01 × 1 0 5 .
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