Determine the product. Write your answer in scientific notation. [tex]$\left(2 \times 10^5\right)\left(5 \times 10^2\right)$[/tex]
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 = 1 \[ \times \] 10^1 \[ \times \] 10^7 .
Express in scientific notation: 1 \[ \times \] 10^8 .
Explanation
Understanding the problem We are asked to determine the product of (2 \[ \times \] 10^5) and (5 \[ \times \] 10^2) and express the result in scientific notation. Scientific notation requires the form a \[ \times \] 10^b where 1 ≤ a < 10 and b is an integer.
Multiplying the coefficients and powers of 10 To find the product, we multiply the coefficients and add the exponents of the powers of 10. 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 the result 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 10 \[ \times \] 10^7 as 1 \[ \times \] 10^1 \[ \times \] 10^7 = 1 \[ \times \] 10^{1+7} = 1 \[ \times \] 10^8 .
Final Answer Therefore, the product in scientific notation is 1 \[ \times \] 10^8 .
Examples
Scientific notation is used in many fields, such as physics, astronomy, and engineering, to represent very large or very small numbers. For example, the distance to the sun is approximately 1.5 × 1 0 11 meters, and the size of an atom is approximately 1 × 1 0 − 10 meters. Using scientific notation makes it easier to work with these numbers and compare their magnitudes. This is particularly useful when dealing with measurements or calculations that involve extreme values.
The product of ( 2 × 1 0 5 ) ( 5 × 1 0 2 ) is 1 × 1 0 8 when expressed in scientific notation. This is achieved by multiplying the coefficients and adding the exponents of the powers of 10. Finally, the result is adjusted to fit the scientific notation format.
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