Recall that large (or big) numbers are far away from zero, and
small numbers are close to zero.
Large and small numbers can be positive (to the right of zero) or negative (to the left of zero).
In this section, you'll do lots of work with powers of ten.
Recall that $\,10^3 = 1000\,$ and $\displaystyle\,10^{3} = \frac{1}{10^3} = \frac{1}{1000}\,$.
More generally, $\,10^n\,$ is a ‘1’ followed by $\,n\,$ zeros; and $\,10^{n}\,$ is $\,1\,$ over $\,10^n\,$.
Multiplying by
$\,10^3\,$ is the same as multiplying by
$\,1000\,$.
Multiplying by ten to a positive power makes a number bigger.
Multiplying by
$\,10^{3}\,$ is the same as dividing by
$\,1000\,$.
Multiplying by ten to a negative power makes a number smaller.
Numbers that are very large
(like
$\,B = 23{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000\,$)
and very small
(like
$\,S = 0{\bf.}00000000000000000000000000000000000000000000000006789\,$)
have long, inconvenient representations using standard notation.
Scientific notation gives a
representation for large and small numbers that is much more compact
and easier to work with.
Here are the scientific notation names for the big number
$\,B\,$ and the small number
$\,S\,$ given above:
$B = 2.3 \times 10^{43}$
$S = 6.789 \times 10^{50}$
Notice that there are two ‘parts’ to each number,
separated by the ‘times’ symbol,
‘$\times$’:
Notice that big numbers have ten raised to a positive power,
and small numbers have ten raised to a negative power.
Here are some more examples:
$3.1 \times 10^{5}\,$  is a small positive number (close to zero, to right of zero) 
$3.1 \times 10^{5}\,$  is a small negative number (close to zero, to left of zero) 
$3.1 \times 10^{5}\,$  is a big positive number (far from zero, to right of zero) 
$3.1 \times 10^{5}\,$  is a big negative number (far from zero, to left of zero) 
Usually, in algebra and beyond, the
‘$\,\times\,$’ symbol is not used for multiplication,
because
it can too easily be confused with the variable $\,x\,$.
Scientific notation is the exception to this rule!
The precise definition of scientific notation follows.
Recall that the integers are the numbers:
$\,\ldots, 3, 2, 1, 0, 1, 2, 3, \ldots$
Changing a number from scientific notation back to standard notation
is a repeated application of the following simple rules:
Numbers have lots of different names,
and one of the most common ways to rename a number is to multiply
by one in an appropriate form.
This idea is used to convert a number in standard notation to scientific notation:
you first ‘stretch or shrink’ to a number between $\,1\,$ and $\,10\,$,
and then restore to the original size, as illustrated next:
People don't typically write out all these steps (phew)!
Instead, most people use the following ‘shortcut’ :
Changing a number from standard notation to scientific notation:
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.However, you can check to see if your answer is correct. 
PROBLEM TYPES:
