Express in scientific notation, using the proper number of significant figures.
$\left(1.8 \times 10^{-2}\right) \div\left(9 \times 10^2\right)=$
A. $0.2 \times 10^{-4}$
B. $2.0 \times 10^5$
C. $2.0 \times 10^{-5}$
D. $2 \times 10^{-5}$
Answer (1)
Divide the coefficients: 1.8/9 = 0.2 .
Divide the powers of ten: 1 0 − 2 /1 0 2 = 1 0 − 4 .
Combine the results: 0.2 × 1 0 − 4 .
Express in scientific notation with one significant figure: 2 × 1 0 − 5 .
Explanation
Understanding the Problem We are asked to divide ( 1.8 × 1 0 − 2 ) by ( 9 × 1 0 2 ) and express the result in scientific notation with the correct number of significant figures. The number 1.8 has two significant figures, while the number 9 has only one significant figure. Therefore, the final answer should have only one significant figure.
Dividing the Coefficients First, we divide the coefficients: 1.8 ÷ 9 = 0.2 .
Dividing the Powers of Ten Next, we divide the powers of ten: 1 0 2 1 0 − 2 = 1 0 − 2 − 2 = 1 0 − 4 .
Combining the Results Combining these results, we have 0.2 × 1 0 − 4 . To express this in scientific notation, we need to write the coefficient as a number between 1 and 10. So, we rewrite 0.2 as 2 × 1 0 − 1 .
Expressing in Scientific Notation Therefore, 0.2 × 1 0 − 4 = ( 2 × 1 0 − 1 ) × 1 0 − 4 = 2 × 1 0 − 1 + ( − 4 ) = 2 × 1 0 − 5 . Since we need only one significant figure, the final answer is 2 × 1 0 − 5 .
Examples
Scientific notation is extremely useful in fields like astronomy and chemistry, where dealing with very large or very small numbers is common. For example, the distance to a star might be 4.5 × 1 0 16 meters, and the size of an atom might be 1 × 1 0 − 10 meters. Using scientific notation makes these numbers easier to write and compare. In this problem, we perform a simple division using scientific notation, which is a fundamental skill in many scientific calculations.
