How to Multiply and Divide in Scientific Notation

By Chris Hren, Peter J. Mikulecky

To make it easier to work with extreme numbers, chemists turn to scientific notation, which is a special kind of exponential notation. A major benefit of presenting numbers in scientific notation is that it simplifies common arithmetic operations like multiplication and division.

Multiplying in scientific notation

To multiply two numbers written in scientific notation, you work out the coefficients and the exponents separately. Here’s an example:

CHEM_0101

To solve this problem, you simply follow these steps:

  1. Multiply the coefficients.
    CHEM_0102
  2. Add the exponents of the powers of 10.
    CHEM_0103
  3. Join the new coefficient to the new power of 10 to get the answer.
    CHEM_0104

Dividing in scientific notation

To divide two numbers written in scientific notation, you also work out the coefficients and the exponents separately, but now it involves division and subtraction. Here’s an example:

CHEM_0105

To solve this problem, you follow these steps:

  1. Divide the coefficients.
    CHEM_0106
  2. Subtract the exponent in the denominator from the exponent in the numerator.

    Remember that the denominator is the bottom number and the numerator is the top number.

    CHEM_0107

  3. Join the new coefficient to the new power of 10.
    CHEM_0108

Okay, time for a few practice questions.

Practice questions

  1. Multiply the following:
    CHEM_0109
  2. Using scientific notation, divide the following:
    CHEM_0110

Answers and explanations

  1. The correct answer is
    CHEM_0111
    First, multiply the coefficients:
    CHEM_0112
    Then multiply the powers of 10 by adding the exponents:
    CHEM_0113
    The raw calculation yields
    CHEM_0114
    which converts to the given answer when you express it in scientific notation:
    CHEM_0115
  2. The correct answer is
    CHEM_0116
    First, convert each number to scientific notation:
    CHEM_0117
    Then divide the coefficients:
    CHEM_0118
    Next, subtract the exponent on the denominator from the exponent of the numerator to get the new power of 10:
    CHEM_0119
    Join the new coefficient with the new power:
    CHEM_0120
    Finally, express gratitude that the answer is already conveniently expressed in scientific notation.