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Integer Exponents

Repeated addition leads to multiplication. For example,

2+2+2+2+2 = 5(2)
Similarly, repeated multiplication leads to exponentiation. For example,
2*2*2*2*2 = 2^5
Here, 5 is the exponent and 2 is the base. Notice that 2^5 is not the same as 5^2 because the first is 32 while the second is 25. Using this understanding of exponentiation, we have the following definitions and rules:
Definitions:
a^0 = 1   (if a isn't zero)
a^1 = a    
a^n = a...a   (for n a positive integer)
a^{-1} = 1/a    
a^{-n} = 1/a^n    
Rules for Exponents:
a^m a^n = a^{m+n}   multiplying powers
(a^m/a^n) = a^{m-n}   dividing powers
(a^m)^n = a^{mn}   raising a power to a power
(ab)^n = a^n b^n   power of a product
(a/b)^n = a^n/b^n   power of a quotient

Notice how the rules for exponents follow from our definition: for example, when we multiply powers we have
a^n a^m = a..a (n factors) a..a (m factors) = a..a (n+m factors) = a^{n+m}
Go through the remaining rules to see how you can derive them from the definition. Yes, now. Get out some paper and write them down.

Note that while these definitions and rules are largely intuitive, there are a couple of places where this may not be the case:

Be careful using the following
ab^n = a(b^n)   (Notice that   ab^n ne (ab)^n   )
-b^n = -(b^n)   (Notice that   -b^n ne (-b)^n   )
-a b^n    
and watch out for this common mistake
(a+b)^n ne a^n + b^n   (The power of a sum is not equal to the sum of the powers)
Examples
As you go through these, be sure that you can explain each step of each example.
  1. 3b(2b)^3 (b^{-1})=3b^1 2^3 b^3 b^{-1}=3(8)b^{3+(-1)+1}=24b^3
  2. y^4(x^3 y^{-2})^2/(2x^{-1})=y^4 x^6 y^{-4}/(2x^{-1})=y^{4-4}x^{6-(-1)}/2=y^0 x^7/2=x^7/2
  3. (2^{-3}/L)^{-2}=2^{(-3)(-2)}/L^{-2}=2^6/L^{-2}=64L^2
  4. 5(2s+1)^4(s+3)^{-2}/(2s+1)=5(2s+1)^{4-1}/(s+3)^2=5(2s+1)^3/(s+3)^2
Practice

Before doing these practice problems, make sure that you have explained each step in the examples above. Then work the practice problems until you are sure you understand them. You can get more practice in the section test at the end of this section.

Note that you can get new practice problems by clicking the "Refresh" button at the bottom of the practice set.

  1. Use the rules of exponents to simplify the following expression into the form
    c   x ^p y ^q

       
    a x ^b ( x ^c y ^d ) ^o   =   x   y  
    (Enter answers, then click: . Answer message: )
  2. Next, use the rules of exponents to simplify this next expression into the same form,
    c   x ^p y ^q

       
    ( g x ^h y ^j ) ^k   ( x ^m y ^n ) ^p   =   x   y  
    (Enter answers, then click: . Answer message: )
  3. Another one! With a fraction this time: again use rules of exponents to simplify the following expression into the same form as before,
    c   x ^p y ^q

    y ^q   ( x ^r y ^s ) ^t
    		  =   x y
       
    ( u x ^v y ^w ) ^z
    (Enter answers, then click: . Answer message: )
For more practice, click: .
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precal: 1.1 - int exponents
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©2003-2007 Gavin LaRose, Pat Shure / University of Michigan Math Dept. / Regents of the University of Michigan