Interactive Algebra Review

(Based on the algebra reviews in Calculus Applied to the Real World, , and Finite Mathematics and Calculus Applied to the Real World and )

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2. Integer Exponents

Positive Exponents

If a is any real number and n is any positive integer, then by an we mean the quantity

thus, a1 = a, a2 = a.a, a5 = a.a.a.a.a

Here are some examples with actual numbers:

In the expression an the number n is called the exponent, and the number a is called the base. The following rules show how to combine such expressions.

Exponent Identities
RuleQuick Example
(a)
aman=am+n
2322  =   25   =   32
(b)
am

an
 = am-n   if m > n and a 0
43

42
 = 43-2  =   41  =  4
(c)
(an)m = anm
(32)2 = 34   =   81
(d)
(ab)n = anbn
(4.2)2 = 4222   =   64
(e)
a

b
 n
 
 		 =
an

bn
4

3
 2
 
 		 =
42

32
 =
16

9

Caution
  • In identities (a) and (b), the bases of the expressions must be the same. For example, rule (a) gives 3234 = 36, but does not apply to 3242.
  • People sometimes invent their own identities, such as am + an = am+n, which is wrong! (If you don't believe this, try it out with a = m = n = 1.) If you wind up with something like 23 + 24, you are stuck with it -- there are no identities around to simplify it further.

Fill in the blanks and press "Check."

(a)
(-2)4(-2)2
=
(-2)
 		 =
(b)
75

73
=
7
 		 =
(c)
x9

(x2)3
=
x
 
(d)
x4y5

(x y2)2
=
x
 		y
 

Negative and Zero Exponents

It turns out to be very useful to allow ourselves to use exponents that are not positive integers. These are dealt with by the following definition.

Negative and Zero Exponents

If a is any real number other than zero and n is any positive integer, then we define

    a-n=
    1

    an
    		 =
    1

    a.a. . . . .a
    		  (n times)

If a is any real number other than zero, then we define

    a0 = 1.
Examples
4-3=
1

43
 =
1

64
 
1,000,0000= 1
 
10-5=
(-2)-4=
x-4

x2
 =
1

x
 
x4

(x-2)3
 = x
 
x4y-2

(x y2)-2
 = x
 		y
 

Simplify each of the following, and express the answer using no negative exponents. (Fill in the blanks and press "Check." Use formula format, for example x^2/y^4 or x^2*y^4)

(a)
x-4y3

x-1y-2
=
(b)
x2

y2 x
 -3
 
 
=

Now go over the examples and try some of the exercises in Section 2 of the Algebra Review of Calculus Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.

Alternatively, press "next" button on the sidebar to go on to the next topic.

Last Updated: November, 1998
Copyright © 1998 Stefan Waner and Steven R. Costenoble