Multiplying and dividing numbers in scientific notation involves straightforward rules for handling both the numerical coefficients and their corresponding exponents.
Multiplying Scientific Notation
When you need to multiply two numbers expressed in scientific notation, follow these two key steps:
- Multiply the Coefficients: Take the numerical parts (the coefficients) of each scientific notation number and multiply them together.
- Add the Exponents: Add the exponents of the base 10 together.
After performing these operations, the result will be in the form of a coefficient multiplied by 10 raised to the new exponent. It is crucial to then adjust this result to ensure it is in proper scientific notation, meaning the coefficient must be a number greater than or equal to 1 and less than 10.
Example: Multiplication
Let's multiply (3.0 × 10^4) by (2.0 × 10^5):
- Multiply coefficients:
3.0 × 2.0 = 6.0 - Add exponents:
4 + 5 = 9 - Initial result:
6.0 × 10^9
In this case, the coefficient 6.0 is already between 1 and 10, so no further adjustment is needed.
Consider another example: (5.0 × 10^3) × (8.0 × 10^2):
- Multiply coefficients:
5.0 × 8.0 = 40.0 - Add exponents:
3 + 2 = 5 - Initial result:
40.0 × 10^5
Here, 40.0 is not between 1 and 10. To convert 40.0 into scientific notation, move the decimal one place to the left to get 4.0. Since you made the coefficient smaller, you must increase the exponent by one: 4.0 × 10^(5+1) = 4.0 × 10^6.
Dividing Scientific Notation
To divide numbers in scientific notation, you'll perform similar operations, but with division for the coefficients and subtraction for the exponents:
- Divide the Coefficients: Divide the numerical part of the numerator by the numerical part of the denominator.
- Subtract the Exponents: Subtract the exponent of the denominator's base 10 from the exponent of the numerator's base 10.
Just like with multiplication, the final answer must be converted back into proper scientific notation if the resulting coefficient is not between 1 and 10.
Example: Division
Let's divide (9.0 × 10^8) by (3.0 × 10^2):
- Divide coefficients:
9.0 ÷ 3.0 = 3.0 - Subtract exponents:
8 - 2 = 6 - Initial result:
3.0 × 10^6
The coefficient 3.0 is already in the correct range, so no adjustment is needed.
Consider another example: (2.0 × 10^4) ÷ (8.0 × 10^2):
- Divide coefficients:
2.0 ÷ 8.0 = 0.25 - Subtract exponents:
4 - 2 = 2 - Initial result:
0.25 × 10^2
Here, 0.25 is not between 1 and 10. To convert 0.25 into scientific notation, move the decimal one place to the right to get 2.5. Since you made the coefficient larger, you must decrease the exponent by one: 2.5 × 10^(2-1) = 2.5 × 10^1.
Summary of Rules
The principles for multiplying and dividing scientific notation can be summarized as follows:
| Operation | Coefficients | Exponents |
|---|---|---|
| Multiply | Multiply | Add |
| Divide | Divide | Subtract |
In both cases, always ensure the final coefficient is between 1 (inclusive) and 10 (exclusive), adjusting the exponent accordingly. For more in-depth practice and understanding, you can explore resources on scientific notation operations.
