- Apply the property of scientific notation multiplication: \((a \times 10^n) \times (b \times 10^m) = (a \times b) \times 10^{n+m}\).
- Multiply the coefficients: \(2.3 \times 3.7 = 8.51\).
- Add the exponents: \(10^5 \times 10^2 = 10^7\).
- Combine to find the final result: \(\boxed{8.51 \times 10^7}\).
Explanation
- Analyzing the Expression
We are given the product of two numbers in scientific notation: \((2.3 \times 10^5) \times (3.7 \times 10^2)\). To solve this, we can apply the properties of exponents and the associative property of multiplication, which allows us to multiply the numerical coefficients and the powers of 10 separately. The rule for multiplying expressions in scientific notation is \((a \times 10^n) \times (b \times 10^m) = (a \times b) \times 10^{n+m}\).
- Performing the Multiplication
First, we multiply the coefficients: \(2.3 \times 3.7\). Using multiplication, we find that \(2.3 \times 3.7 = 8.51\). Next, we multiply the powers of 10 by adding their exponents: \(10^5 \times 10^2 = 10^{5+2} = 10^7\). Combining these results, we get \(8.51 \times 10^7\).
- Verification and Conclusion
The standard form for scientific notation is \(c \times 10^n\), where \(1 \leq c < 10\). Our calculated value is \(8.51 \times 10^7\). Since \(8.51\) is between \(1\) and \(10\), this value is already in standard scientific notation. Comparing our result with the given options, we find it matches the fifth option.
Examples
Scientific notation is a common way to express very large or very small numbers, which is essential in fields like astronomy, physics, and chemistry. For example, when calculating the total distance light travels in a year, or the number of atoms in a sample, scientists use this notation to simplify calculations and avoid writing down many trailing or leading zeros. Mastering this technique helps in handling complex data sets and understanding scales of magnitude efficiently.