- Multiply the coefficients: \(5.5 \times 2.2 = 12.1\).
- Combine the powers of ten by adding exponents: \(10^4 \times 10^{-8} = 10^{-4}\).
- Adjust the result \(12.1 \times 10^{-4}\) to standard scientific notation by moving the decimal, resulting in \(1.21 \times 10^{-3}\).
- Final Answer: \(\boxed{1.21 \times 10^{-3}}\)
Explanation
- Problem Analysis and Method Selection
To compute the product of two numbers in scientific notation, we follow the property \((a \times 10^n) \times (b \times 10^m) = (a \times b) \times 10^{n+m}\). Here, the given values are \(5.5 \times 10^4\) and \(2.2 \times 10^{-8}\). We identify \(a = 5.5\), \(b = 2.2\), \(n = 4\), and \(m = -8\).
- Calculation of Coefficients and Exponents
First, we multiply the coefficients: \(5.5 \times 2.2 = 12.1\). Next, we sum the exponents: \(10^4 \times 10^{-8} = 10^{4 + (-8)} = 10^{-4}\). Combining these gives us the expression \(12.1 \times 10^{-4}\).
- Converting to Standard Scientific Notation
Scientific notation requires the coefficient to be in the range \(1 \leq |a| < 10\). Our current result is \(12.1 \times 10^{-4}\). To adjust it, we write \(12.1\) as \(1.21 \times 10^1\). Substituting this back into the expression, we get: \((1.21 \times 10^1) \times 10^{-4} = 1.21 \times 10^{1-4} = 1.21 \times 10^{-3}\). This final result matches the third option provided.
Examples
Scientific notation is crucial for scientists and engineers when working with extremely small or large measurements. For example, when calculating the mass of an electron or the distance between galaxies, working with decimals like \(0.000000000000000000000000000000911\) kg is cumbersome. Converting these to scientific notation, such as \(9.11 \times 10^{-31}\) kg, makes multiplication, division, and comparison much faster and reduces the likelihood of errors when handling tiny values in physical calculations.