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 $4.56\times10,000,000$. So it can be written as $4.56\times10^7$, and $4.56\times10^7$ is the number in scientific notation. 

The number $0.00006772$ is a small number, and it is $6.772\times 0.00001=6.772\times \dfrac{1}{100000}=6.772\times \dfrac{1}{10^5}=6.772\times 10^{-5}$, and $6.772\times 10^{-5}$ 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

$a\times 10^n$.

Here the absolute value of number $|a|$ must be greater than (or equal) to 1 and less than 10,  $1\le a<10$, 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)     $-225,000=-2.25\times 10^5$

b)     $0.0155=1.55\times 10^{-2}$

c)     $56.7\times 10^8=5.67\times 10\times 10^8=5.67\times 10^9$  (Note that for scientific notation, the number $a$ must be between 1 and 10)

d)     $-0.88\times 10^{-4}=-8.8\times 10^{-1}\times 10^{-4}=8.8\times 10^{-5} $  (Note that for scientific notation, the number $a$ must be between 1 and 10)

 

Multiply Two Numbers in Scientific Notation:

$(a\times 10^m)(b\times 10^n)=(a\cdot b)\times 10^{m+n}$

Divide Two Numbers in Scientific Notation:

$\dfrac{a\times 10^m}{b\times 10^n}=\dfrac{a}{b}\times 10^{m-n}$

 

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

a)     $(6.2\times 10^8)(3.0\times 10^7=(6.2)( 3.0)\times 10^{8+7}=18.6\times 10^{15}$

         $=1.86\times 10\times 10^{15}= 1.86\times 10^{16}$

b)     $\dfrac{4\times 10^{5}}{8\times 10^{-3}}=\dfrac{4}{8}\times 10^{5-(-3)}=0.5\times 10^8$

         $=5\times 10^{-1}\times 10^{8}=5\times 10^7$

 

 

Practice Problems: 

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

(a)   $(1.5\times 10^6)(2.7\times 10^7$

(b)   $(-6.6\times 10^{-6})(7\times 10^9$ 

(c)   $\dfrac{5.1\times 10^{5}}{1.7\times 10^{2}}$

(d)   $\dfrac{2.6\times 10^{4}}{-5.2\times 10^{-2}}$    

 

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

 

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