Exponent Properties
1. Product of like bases:
a m a n = a m+ n
To multiply powers with the same base, add the exponents and keep the common base.
Example: x 5 x 3 = x 5+ 3 = x 8
am
m− n
n = a
a
To divide powers with the same base, subtract the exponents and keep the common base.
x5
Example: 3 = x 5−3 = x 2
x
2. Quotient of like bases:
3. Power to a power:
(a m ) n = a mn
To raise a power to a power, keep the base and multiply the exponents.
( )
Example: x 5
3
= x 5*3 = x 15
4. Product to a power:
(ab) m = a mb m
To raise a product to a power, raise each factor to the power.
(
Example: x 4 y 5
)
3
= x 12 y 15
n
n
a a
5. Quotient to a power
  = n
b b
To raise a quotient to a power, raise the numerator and the denominator to the power.
 x3
Example:  2
y



4
=
x 12
y8
6. Zero Exponent:
a0 = 1
Any number raised to the zero power is equal to “1”.
Example: (8 x 4 ) 0 = 1
1
1
or
= an
n
−n
a
a
Negative exponents indicate reciprocation, with the exponent of the reciprocal becoming
positive. You may want to think of it this way: unhappy (negative) exponents will become
happy (positive) by having the base/exponent pair “switch floors”!
1
1
4
Example: 8 − 2 = 2 =
or
= 4x3
−3
64
x
8
7. Negative exponent:
a −n =
Common Algebraic Errors
Error
Reason/Correct/Justification/Example
2
2
 0 and  2
0
0
Division by zero is undefined!
32  9
32  9 ,
x 
x 
2 3
2 3
 x5
a
a a
 
bc b c
1
 x 2  x 3
2
3
x x
 a  x  1   ax  a
x2  a2  x  a
xa  x  a
 x  a
 x n  a n and
n
 9 Watch parenthesis!
 x2 x2 x2  x6
 x  a
 x2  a2
2
2
1
1
1 1

  2
2 11 1 1
A more complex version of the previous
error.
a  bx a bx
bx
 
 1
a
a a
a
Beware of incorrect canceling!
 a  x  1   ax  a
Make sure you distribute the “-“!
a  bx
 1  bx
a
 x  a
 3 
n
xa  n x  n a
  x  a  x  a   x 2  2ax  a 2
2
5  25  32  42  32  42  3  4  7
See previous error.
More general versions of previous three
errors.
2  x  1  2  x 2  2 x  1  2 x 2  4 x  2
2
2  x  1   2 x  2 
2
 2 x  2
2
2
2
 2  x  1
 2 x  2
 4 x2  8x  4
Square first then distribute!
See the previous example. You can not
factor out a constant if there is a power on
the parethesis!
2
1
 x2  a2   x2  a2
a
ab

b c
 
c
a
  ac
b 
c
b
 x2  a2    x2  a 2  2
Now see the previous error.
a
 
a
1
 a  c  ac
       
 b   b   1  b  b
   
c c
a a
   
 b    b    a  1   a
  
c
 c   b  c  bc
 
1
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins