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.

Deal with numbers and 10's separately  

Multiply numbers  

Use product rule on 10's and add exponents  

Our Solution 
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.

Deal with numbers and 10's separately  

Divide numbers  

Use quotient rule to subtract exponents, be careful with negatives!  





Our Solution 
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.

Use power rule to deal with numbers and 10's separately  

Evaluate  

Multiply exponents 


Our Solution 

Deal with numbers and 10's separately  

Multiply numbers  

Convert this number into scientific notation  

Use product rule, add exponents, using from conversion  

Our Solution 

Deal with numbers and 10's separately  

Divide numbers  

Change this number to scientific notation  

Use product and quotient rule, using from the conversion  


Be careful with signs: 

Our Solution 