Exponents are the mathematician's shorthand.

In general, the format for using exponents is:

(base)exponent

where the exponent tells you how many of the base are being multiplied together.

Consider:    2 2 2  is the same as 23, since there are
three 2's being multiplied together.

Likewise,   5 5 5 5 = 54,   because there are
four 5's being multiplied together.
 


Exponents are also referred to as "powers".
 For example, 23 can be read as "two cubed" or as "two raised to the third power".

 

Exponents of Negative Values

When we multiply negative numbers together, we must utilize parentheses to switch to exponent notation.
                                (-3)(-3)(-3)(-3)(-3)(-3) = (-3)6

           BEWARE!!   -36 is NOT the same as (-3)6
The missing parentheses mean that -36 will multiply six 3's together first (by order of operations), and then take the negative of that answer.

(-3)6 = 729     but      -36 = -729
 so be careful with negative values and exponents !

 

Note:  Even powers of negative numbers allow for the negative values to be arranged in pairs.  This pairing guarantees that the answer will always be positive.  

(-5)6 = (-5)•(-5)  •   (-5)•(-5)   •   (-5)•(-5)   ← All pairs.
  =     25       •       25       •       25
  =  15625   (a positive answer)

Odd powers of negative numbers, however, always leave one factor of the negative number not paired.  This one lone negative term guarantees that the answer will always be negative.
 

(-5)5 = (-5)•(-5)  •   (-5)•(-5)   •   (-5)  ←One lone, un-paired, negative.
  =     25       •       25       •   (-5)
  =  -3125   (a negative answer)


 

Zero Exponents

The number zero may be used as an exponent. 
The value of any expression raised to the zero power is 1.
(Except zero raised to the zero power is undefined.) 

Base0

Value

20  =

1

(-6)0  =

1

40  =

1

-80  =

-1
Raise to the zero power first:  80=1
then take the negative.

00 =

undefined

 

Negative Exponents

Negative numbers as exponents have a special meaning. 
The rule is as follows:

base negative exponent  = 

 

For example:

Negative Exponent

Positive Exponent

4-1  =

7-3  =

(-5)-2  =


 

 Exponents and Units

When working with units and exponents (or powers), remember to adjust the units appropriately.

(36 ft)3    

 = (36 ft) • (36 ft) • (36 ft)
  = (36 • 36 • 36) (ft • ft • ft)
  = 46656 ft3

 

Exponents can be very useful for evaluating expressions.  It is also useful to learn how to use your calculator when working with exponents.
 Instructions for using the TI-83+/84+ graphing calculator with exponents.