[PDF] 3 Integer Exponents

944_63IntegerExps.pdf

3 Integer Exponents

For a non-negative integern:

The notationanrepresentsamultiplied by itselfntimes. ex:53= 555. Ifa6= 0, thena0= 1becauseamultiplied by itself zero times is a product that has no terms, and a product that has no terms equals1. Note:00is not a number; it is an indeterminate form that will be studied in calculus. Multiplying exponents:When multiplying powers of the same base, add the exponents, because: a nam=aa


a|{z} ntimesaa


a|{z} mtimes=aa


aaa


a|{z} n+mtimes=an+m When multiplying exponential expressions of different bases but of the same power, multiply by the bases together and raise it to the exponent, because: a nbn=aa


a|{z} ntimesbb


b|{z} ntimes=abab


ab|{z} ntimes= (ab)n Powers of Powers:When raising a power to another power, multiply the exponents, be- cause: (an)m=anan


an|{z} mtimes=aaaa


aaaaaa


aa|{z} nmtimes=anm Negative exponentsare a shorthand for a power of the reciprocal, and only make sense if the base is not zero. That is, forn >0anda6= 0, a n=a1
n=1a  n =1a n:

Note thata1=1a

: Dividing exponents:When dividing powers of the same base, subtract the exponents be- cause of the meaning of negative exponents (above): a nam=ana m=an1a m =anam=anm: If you have different bases to different powers, sometimes you can combine by factoring.

See below.

Example 1

63
67= 63+7= 610 1 24
27= 24+7= 211:Notice that211can arise other ways; for example211= 28+3= 28
23 (460)2= 4602= 4120  1343
=17

3= 73

9692= 962= 94  343 5= 34(5)= 31= 3 4353= (45)3= 203 25
86= 25
(23)6= 25
218= 223 32
94= 32
(32)4= 32
(38) = 36 (35)2
79= (5
7)2
79= 52
711 123
184= (2
6)3
(3
6)4= 23
63
34
64= 23
34
67= 3
63
67= 3
610

3.1 Practice Problems

Use the rules of exponents to find an expression equivalent to the following:

1.z4
z5

2.(93)4

3.(22)4

4. q5q 2

5.(4)3
53

3.2 Solutions

1.z12.912= 3243.284.q35.(20)3=(203) =203

In each case there are other correct solutions.

2