The distance between two points is $2.54 \times 10^{-2} \text{ m}$. Express this distance in km in standard form.
The given distance between two points is \(2.54 \times 10^{-2} \text{ m}\). We need to convert this distance into kilometers (\(\text{km}\)) and express the result in scientific notation (standard form). Scientific notation is a way to express numbers in the form \(a \times 10^n\), where \(1 \leq |a| < 10\) and \(n\) is an integer.
To convert meters to kilometers, we use the fact that \(1 \text{ km} = 10^3 \text{ m}\). This implies that \(1 \text{ m} = 10^{-3} \text{ km}\).
To find the distance in kilometers, we multiply the given distance in meters by the conversion factor:
\[2.54 \times 10^{-2} \text{ m} \times 10^{-3} \text{ km/m}\]
Using the laws of exponents, specifically \(10^a \times 10^b = 10^{a+b}\), we combine the powers of 10:
\[2.54 \times 10^{-2 + (-3)} \text{ km} = 2.54 \times 10^{-5} \text{ km}\]
The result \(2.54 \times 10^{-5}\) is already in the required standard form \(a \times 10^n\), as \(1 \leq 2.54 < 10\) and \(-5\) is an integer. Thus, the distance in kilometers is \(2.54 \times 10^{-5} \text{ km}\).
In scientific research, especially when dealing with microscopy or nanotechnology, distances are often measured in small units like meters or micrometers. When comparing these to larger-scale measurements such as kilometers, researchers must convert their data consistently. For example, if a researcher measures the width of a microscopic particle to be \(2.54 \times 10^{-2} \text{ m}\), converting this to kilometers helps provide context relative to macroscopic distances, ensuring accuracy in scaling and comparative analysis. This fundamental skill is essential for maintaining precision across different fields of science.