Compare and contrast adding/subtracting numbers in scientific notation with multiplying/dividing numbers in scientific notation

Answer 1

Adding/subtracting and multiplying/dividing numbers in scientific notation involve different processes and considerations. Here's a detailed comparison:

Adding/Subtracting Numbers in Scientific Notation:

Process:

1. Align the Exponents: Ensure that both numbers have the same exponent. If the exponents are different, adjust one number by converting it to have the same exponent as the other.
2. Add/Subtract the Coefficients: Once the exponents are equal, add or subtract the coefficients (the numbers in front).
3. Keep the Exponent: The exponent remains the same as that of the adjusted numbers.
4. Normalize: If necessary, adjust the result so that the coefficient is between 1 and 10 and the exponent is adjusted accordingly.

Example:

• Add \(3.0 \times 10^5\) and \(2.5 \times 10^6\):
  1. Convert \(2.5 \times 10^6\) to \(25.0 \times 10^5\).
  2. Now you have \(3.0 \times 10^5 + 25.0 \times 10^5 = (3.0 + 25.0) \times 10^5 = 28.0 \times 10^5\).
  3. Normalize: \(2.8 \times 10^6\).

Multiplying/Dividing Numbers in Scientific Notation:

Process:

1. Multiply/Divide the Coefficients: Perform the multiplication or division on the coefficients.
2. Add/Subtract the Exponents: For multiplication, add the exponents. For division, subtract the exponents.
3. Normalize: Ensure the result has a coefficient between 1 and 10 and adjust the exponent if necessary.

Example:

• Multiply \(3.0 \times 10^5\) and \(2.0 \times 10^6\):

  1. Multiply the coefficients: \(3.0 \times 2.0 = 6.0\).
  2. Add the exponents: \(5 + 6 = 11\).
  3. Result: \(6.0 \times 10^{11}\) (already normalized).

• Divide \(4.0 \times 10^8\) by \(2.0 \times 10^3\):

  1. Divide the coefficients: \(4.0 / 2.0 = 2.0\).
  2. Subtract the exponents: \(8 - 3 = 5\).
  3. Result: \(2.0 \times 10^5\) (already normalized).

Key Differences:

1. Alignment of Exponents: In addition and subtraction, it’s necessary to align the exponents first, while in multiplication and division, you manipulate the exponents directly based on the operation (addition for multiplication, subtraction for division).

2. Operation on Exponents: For addition and subtraction, the exponents must be the same to combine coefficients, while in multiplication and division, you combine coefficients and update exponents straightforwardly.

3. Complexity and Steps: Addition and subtraction of numbers in scientific notation may require additional steps to equalize the exponents, while multiplication and division typically proceed more directly.

Conclusion:

In summary, adding and subtracting require careful handling of exponents to ensure they are the same before performing operations on the coefficients. In contrast, multiplying and dividing directly work with the coefficients and manipulate the exponents based on the operations being performed. This fundamental difference underscores the need for different approaches to these operations in scientific notation.