- Express \(0.000271\) as \(2.71 \times 10^{-4}\) to start the conversion process.
- Combine the powers of 10 using the exponent addition rule: \(10^{-4} \times 10^6 = 10^{-4+6} = 10^2\).
- State the final simplified scientific notation as \(2.71 \times 10^2\).
- Final Answer: \(\boxed{2.71 \times 10^2}\)
Explanation
- Problem Analysis
To represent a number in scientific notation, it must be written in the form \(a \times 10^n\), where the coefficient \(a\) satisfies \(1 \leq |a| < 10\) and \(n\) is an integer. We are given the expression \(0.000271 \times 10^6\) and need to identify its standard scientific notation form. We will start by converting the decimal part into scientific notation and then combine it with the power of 10.
- Converting the Decimal Coefficient
First, we convert \(0.000271\) into scientific notation. To move the decimal point to the right to obtain a coefficient between 1 and 10, we count the number of places shifted. Moving the decimal point 4 places to the right gives us \(2.71\). Since we moved the decimal 4 places to the right, we multiply by \(10^{-4}\). Thus, \(0.000271 = 2.71 \times 10^{-4}\).
- Simplification and Final Result
Now we substitute this back into the original expression:
\[(2.71 \times 10^{-4}) \times 10^6\]
Using the laws of exponents, specifically the rule \(10^m \times 10^n = 10^{m+n}\), we can combine the powers of 10:
\[2.71 \times 10^{-4+6} = 2.71 \times 10^2\]
This result is in the form \(a \times 10^n\) where \(a = 2.71\) (which satisfies \(1 \leq 2.71 < 10\)) and \(n = 2\). Comparing this to the given options, we find that this matches the correct form.
Examples
Scientific notation is widely used in fields like astronomy and chemistry to handle very large or very small numbers. For instance, the mass of a single dust particle might be \(0.000000000753\) kilograms. Instead of writing all those zeros, a scientist writes \(7.53 \times 10^{-10}\) kg, which makes calculations and comparisons much easier. This format allows researchers to focus on significant digits and powers of ten, minimizing the risk of errors when writing or reading extremely small or large measurements.