Scientific notation can be used to express very large and very small numbers by multiplying a decimal number by a power of 10. By very large number, we mean a number containing several digits to the left of the decimal, such as 23,000,000 (23 million). By small number, we mean a number with zero in the ones place, with several decimal digits behind it, such as 0.000023 (23 millionths).
Generally, numbers written in scientific notation can be expressed as a • 10^n, where a is a decimal number, and 10^n represents a power of ten.
In proper scientific notation, there are a couple of restrictions to what “a” is allowed to be. “a” can only contain one nonzero digit to the left of the decimal. Here are some examples of numbers that might look like they are in scientific notation, but they violate this rule for what “a” can be:

A zero to the left of the decimal is not allowed  

More than 1 digit to the left of the decimal is not allowed 
In cases where there is a zero or more than one digit to the left of the decimal, we can move the decimal and adjust the exponent. Let’s change our previous examples into proper scientific notation:

A zero to the left of the decimal is not allowed  

Decimal moved right 1 place, exponent decreased by 1  




More than one digit to the left of the decimal is not allowed  

Decimal moved left 1 place, exponent increased by 1 
How does this help us add and subtract numbers in scientific notation? We can easily add and subtract numbers in scientific notation if they are expressed using the same power of 10. This is illustrated in the following example:




Add the decimal numbers together, keep the power of 10  

Our Solution 
We can confirm this is a valid process by showing the addition in standard notation:




Writing in standard notation  

Adding the numbers  

Our solution in scientific notation 
What if we wish to add or subtract numbers in scientific notation that have unlike powers of 10? We can use our handy trick of moving the decimal and adjusting the exponent to express the numbers with a common power of 10. This is illustrated in the following examples:




Rewrite to have a common power of 10  

Group the decimal numbers, keep power of 10  

Subtract 21 from 4.1  

Our Solution 
Note that it does not matter which number you choose to rewrite. We would get the same result if we chose to rewrite the other number, although we might give ourselves more work to do. Let’s go through the same example, but in a different way:




Rewrite one number to have a common power of 10  

Subtract decimal numbers, keep power of 10  

Our Solution 
As you can see in the second example, we needed to make a final adjustment to our solution, because 16.9 x 10^3 is not in proper scientific notation. No matter which number you decide to rewrite, always be sure to make any necessary movements to the decimal, and increase or decrease your exponent accordingly.
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html