In this section, we will learn how to operate (multiply and divide) numbers in scientific notation.  

 

There is a mathematical scientific notation which is very useful for writing very large and very small numbers.

For example:

The number 45,600,000 is a large number,  and it is 4.56×10,000,000. So it can be written as 4.56×107, and 4.56×107 is the number in scientific notation. 

The number 0.00006772 is a small number, and it is 6.772×0.00001=6.772×1100000=6.772×1105=6.772×10−5, and 6.772×10−5 is the number in scientific notation .     (Note that in the last step, we used the definition of negative power mentioned in the section 3.1) 

 

Scientific Notation:

A number is said to be written in scientific notation if it is written as

a×10n.

Here the absolute value of number |a| must be greater than (or equal) to 1 and less than 10,  1≤a<10, and the decimal number a  is followed by multiplication by a power of 10.

 

Example: The given numbers are not in scientific notation. Modify each so that your answer is in scientific notation. 

a)     −225,000=−2.25×105

b)     0.0155=1.55×10−2

c)     56.7×108=5.67×10×108=5.67×109  (Note that for scientific notation, the number a must be between 1 and 10)

d)     −0.88×10−4=−8.8×10−1×10−4=8.8×10−5  (Note that for scientific notation, the number a must be between 1 and 10)

 

Multiply Two Numbers in Scientific Notation:

(a×10m)(b×10n)=(a⋅b)×10m+n

Divide Two Numbers in Scientific Notation:

a×10mb×10n=ab×10m−n

 

Example: Perform the given operation and write your answer in scientific notation:

a)     (6.2×108)(3.0×107=(6.2)(3.0)×108+7=18.6×1015

         =1.86×10×1015=1.86×1016

b)     4×1058×10−3=48×105−(−3)=0.5×108

         =5×10−1×108=5×107

 

 

Practice Problems: 

1. Perform the given operation and write your answer in scientific notation:

(a)   (1.5×106)(2.7×107

(b)   (−6.6×10−6)(7×109 

(c)   5.1×1051.7×102

(d)   2.6×104−5.2×10−2    

 

Answer Key: 1(a)         (b)          (c)            (d)                     

 

For more detailed explanation, please read: Arithmetic|Algebra Chapter 6.