Interactive Algebra Review

This topic can calso be found in Section 0.2 of Applied Calculus and Finite Mathematics

0.2 Exponents and Radicals

This tutorial: Part A: Integer Exponents
Next tutorial: Part B: Radicals and Rational Exponents

Positive Exponents

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

The number a is called the base and the number n is called the exponent.

Thus, a1 = a,   a2 = a.a,   a5 = a.a.a.a.a

Here are some examples with actual numbers:

The following rules show how to combine such expressions.

Exponent Identities
RuleExample
(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 missing exponents and other numbers and press "Check.". (Raised boxes are exponents.)

Here are some for you to try.

Rational and Exponential Form
An expression of rational form is an expression written as a ratio:
axm

bxn
 where m and n ≥ 0.

Examples: Expressions in Rational Form:
2.1x2

2x4
 ,  
4

7x4
 ,  
-3x4

2.3
 ,  
22

7
but
4x-4

7
 is not in rational form because the exponent of x is negative.

An expression of exponential form is an expression written as axn where n is any exponent (possibly negative or zero).

Examples: Expressions in Exponential Form:
4.1x-3 ,  
2

3
x2 ,  
2

5.1
x-4 ,  
22

7
but
4

7x-4
 is not in exponential form because of the x is in the denominator.

Converting between Rational and Exponential Form

We can use the exponent identities to convert between the two forms just described:

Rational FormExponential Form
4

7x4
4

7
x-4
2.1x2

2x4
1.05x-2
4x4

7
4

7
x4
3

2x5
 
2

7x-1
 
3.2x-6
 
-4

3
 x-1
 

Simplify each of the following, and express the answer using no negative exponents.

You should go over Part B in the next tutorial before trying the examples and exercises in Section 0.2 of the Algebra Review of Applied Calculus and Finite Mathematics and Applied Calculus

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Last Updated: December, 2007
Copyright © 2007 Stefan Waner