eclectic content

exponentiation


[ ]
[ eks' -poh -nehn   tee -aye -shun ] or [ ek - spuh -nehn  shee - aye -shuhn] 

10 ^ 9 
means 'ten to the ninth power'
which means, in turn,
10x10x10x10x10x10x10x10x10.

In computing, the following notation is generally used:
10^9.

You can try this in Google: '10^9=' (Go)
or on a scientific calulator using the 'x ^ y' button.

Q: What is 10 to the first power?
 Answer: 10 (essentially, "one (instance of) Ten", or "Ten by itself"). For comparison, Ten to the second power is 10 x 10.)

Q: What is 10 to the zero power (100)?
 Answer: 1.

 

The elements of an "exponential expression" are

  • a number (a "base"), and
  • a raised number (an "exponent");

the base is "raised to the power of" the exponent.

 

A simple way of thinking about powers of ten is that the exponent indicates the number of zeros:

  • 101 = 1 & one zero = 10
  • 100 = 1 & zero zeroes = 1
  • 102 = 1 & two zeroes = 100

Note: Exponents are somewhat unique in that they focus on "tens"; other Factors (x^y) are slightly different because they don't behave the same way; for example two to the power of 2 (22 = 4) has nothing to do with zeros..

Tip: It may be more useful for you to think of powers of ten in terms of "zero-places" rather than just "zeros"-- this allows you to use the concept for negative exponents (10-1=0.1) and 'scientification notation' (see following section).


Exponentiation in "Scientific notation"

Exponentiation is used in scientific notation to abbreviate large numbers.

For example, instead of writing 123,000,000,000 , you could write 1.23 x 1011 or 1.23 x (10^11)

The 'Exp' button seen in the image above indicates this type of abbreviation. This can be written as

1.23E+11   or    1.23e+11=

A simple way of speaking about this notation is to say "move the decimal over 11 places":

moving the decimal 11 places to the left

Or, if it's a whole number, you could just think of it as adding eleven zeros.
If it's not a whole number, you could think of it as adding eleven zero-places (that is, 1e+11 is 1 followed by eleven zeros; 1.23e+11 is 1 followed by eleven zero-places, with the 2 and the 3 taking the first and second zero-places).

Note that the "exponentiation"/"scientific" notation for the example used on the first part of this page (ten raised to the ninth power) is 1.e+9 .

 

 

 


 10^1  	| 10^3		| 10^6		| 10^9
 10     | 1,000		| 1,000,000	| 1,000,000,000
 ten	| one thousand	| one million	| one billion
 Bytes	| KB		| MB		| GB
 Bytes	| ThB		| MiB		| BB
 in excel:
					=10^9
					1,000,000,000
					=10**9
					10,000,000,000
					(= 10 with 9 zeros? )
 in Google:
	10^9=     1 000 000 000
	10**9=
	10 ** 9 = 1 000 000 000
  

  
Links:

http://www.nyu.edu/pages/mathmol/textbook/scinot.html

http://dl.clackamas.cc.or.us/ch104-02/exponents.htm

 

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