Perform the following operation and express the answer in proper scientific notation. [tex]$\frac{9.0 \times 10^{-5}}{2.0 \times 10^{-8}}=[?] \times 10^{[?]}$[/tex] Enter the coefficient in the green box and the exponent in the yellow one. Coefficient (green) [tex]$\square[/tex] Exponent (yellow)
Answer (2)
To divide the numbers in scientific notation, divide the coefficients and subtract the exponents of the powers of ten. This gives us 4.5 × 1 0 3 . Therefore, the coefficient is 4.5 and the exponent is 3.
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Divide the coefficients: 2.0 9.0 = 4.5 .
Divide the powers of 10: 1 0 − 8 1 0 − 5 = 1 0 3 .
Combine the results: 4.5 × 1 0 3 .
Express the final answer in scientific notation: 4.5 × 1 0 3 .
Explanation
Understanding the Problem We are asked to divide two numbers expressed in scientific notation and express the result in proper scientific notation. Let's break down the problem into smaller, manageable steps.
Dividing the Coefficients First, we divide the coefficients: 2.0 9.0 .
Calculating the Coefficient The result of dividing 9.0 by 2.0 is 4.5.
Dividing the Powers of 10 Next, we divide the powers of 10: 1 0 − 8 1 0 − 5 . When dividing exponential terms with the same base, we subtract the exponents: 1 0 − 5 − ( − 8 ) = 1 0 − 5 + 8 = 1 0 3 .
Combining the Results Now, we combine the results from the coefficient and the exponent: 4.5 × 1 0 3 . This is already in proper scientific notation since the coefficient, 4.5, is between 1 and 10.
Final Answer Therefore, the final answer in scientific notation is 4.5 × 1 0 3 .
Examples
Scientific notation is extremely useful in fields like astronomy and chemistry where you often deal with very large or very small numbers. For example, the distance to the nearest star (other than the Sun) is approximately 4.0 × 1 0 16 meters. Similarly, the size of an atom might be around 1.0 × 1 0 − 10 meters. Performing calculations with these numbers is much easier when they are expressed in scientific notation, preventing errors and simplifying the math.
