Multiplying and Dividing with Scientific Notation
When converting between standard notation and scientific notation it is important to understand how scientific notation works and what it does. Here our main interest is to be able to multiply and divide numbers in scientific notation using exponent properties. The way we do this is to first do the operation with the front number (multiply or divide) then use exponent properties to simplify the 10’s. Scientific notation is the only time where it will be allowed to have negative exponents in our final solution. The negative exponent simply informs us that we are dealing with small numbers.
Here is an example of multiplying two numbers in scientific notation. Pay particular attention to which numbers we multiply first, and how we deal with the powers of 10.
A similar process is used to divide in scientific notation. First, we divide the decimal part of the number, and then apply a property of exponents to easily divide the powers of 10.
Notice that the properties of exponents can always be used in scientific notation. This is because the exponent properties apply when the bases are the same, and the base is always 10 in scientific notation.
Do you think we can use any properties of exponents as shortcuts when we raise a number in scientific notation to an exponent power?
When multiplying and dividing in scientific notation, deal with the non-exponential and exponential (containing the power of 10) separately. Multiply or divide the decimal number part first, then multiply or divide the part with exponents, applying the property of exponents to either add or subtract the exponents.
Often when we multiply or divide in scientific notation the end result is not in scientific notation. We will then have to convert the front number into scientific notation and then combine the 10’s using the product property of exponents and adding the exponents. This is shown in the following examples.
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html