Determine the product. Write your answer in scientific notation.
$\left(2 \times 10^5\right)\left(5 \times 10^2\right)$
Answer (2)
Multiply the coefficients: 2 \[\times\] 5 = 10 .
Multiply the powers of 10: 10^5 \[\times\] 10^2 = 10^{5+2} = 10^7 .
Combine the results: 10 \[\times\] 10^7 .
Express in scientific notation: \boxed{1 \[\times\] 10^8} .
Explanation
Understanding the problem We are given the expression (2 \[\times\] 10^5)(5 \[\times\] 10^2) and asked to determine the product in scientific notation. Scientific notation requires the form a \[\times\] 10^b where 1 ≤ a < 10 and b is an integer.
Multiplying coefficients and powers of 10 To find the product, we multiply the coefficients and the powers of 10 separately. First, multiply the coefficients: 2 \[\times\] 5 = 10 . Next, multiply the powers of 10: 10^5 \[\times\] 10^2 = 10^{5+2} = 10^7 .
Expressing in scientific notation Now, combine the results: (2 \[\times\] 10^5)(5 \[\times\] 10^2) = 10 \[\times\] 10^7 . To express this in scientific notation, we need the coefficient to be between 1 and 10. Since the coefficient is 10, we rewrite it as 10 = 1 \[\times\] 10^1 . Therefore, 10 \[\times\] 10^7 = (1 \[\times\] 10^1) \[\times\] 10^7 = 1 \[\times\] 10^{1+7} = 1 \[\times\] 10^8 .
Final Answer The product in scientific notation is 1 \[\times\] 10^8 .
Examples
Scientific notation is extremely useful in fields like astronomy and physics, where dealing with very large or very small numbers is common. For example, the distance to the nearest star, Proxima Centauri, is approximately 4.017 \[\times\] 10^{13} kilometers. Similarly, the size of an atom is around 1 \[\times\] 10^{-10} meters. Using scientific notation makes these numbers easier to handle and compare.
The product of ( 2 × 1 0 5 ) ( 5 × 1 0 2 ) is calculated by multiplying the coefficients and the powers of ten separately. The final result in scientific notation is 1 × 1 0 8 .
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