P.2.s s
IntegralExponents
Notation
and Scientific
15
c)
Sandyhasonequartofgrassseedandonequartofsand,eachstoredin onegallon containers.Sandypoursa little seedinto the sandandshakeswell. Shethen
poursthe sameamountof the mix backinto the containerof seedso that both
containersagaincontainexactlyone quart.Is theremore sandin Sandy'sseed
or moreseedin Sandy'ssand?Explain.
d)
In each cell of the secondrow of the following table put one of the digits 0 through
9. You may use a digit more than once, but each digit in the secondrow must indicate the number of times that the digit above it appearsin the secondrow.
6 . 2 .t , 0 . 0 . 0 . 1 , 0 . 0 , 0
The percentageofseed ir.rthe sandis equalto the percentageofsarrd in the seed.
i*0
*-i-f-l;-
i--**-l^-"*li-**-
-1
?
]*i;-]
[-3--T;"T;*ru-I
{- - -*i* -"} * -:.+'**-T-**1-***l-**-
I
Integral Exponentsand Scientific Notation
We defined positive integral exponentsin SectionP.1.In this sectionwe will define
negative integral exponents and review the rules for exponents.Then we will see
how exponentsare used in scientific notation to indicate very large and very small
numbers.
Negative Integral Exponents
We use a negative sign in an exponentto representmultiplicative inversesor reciprocals. For negative exponentswe do not allow the base to be zero becausezero does
not have a reciprocal.
i*-*-**-*'*"*..
1:
Definition:Negative
Integral Exponents
: If o i, a nonzero real number and n is a positive integer, th"n o-n :
foa.nF/a I
).
rvatuatingexpressions
that have
negative exponents
Simplify each expressionwithout using a calculator, then check with a calculator.
a . 3 - r . s - z. t o 2 o . / ? \ - '
\l/
^ 6-2
c'
24
Solution
a.3r.5r.lo2
lt
J)-
ll
t 0 0: - . - .
325
100
100 4
753
I
16
Chapter P I sffi
Prerequisites
o(3)'
- 1 + 5 * - 2 + 1 B rF F F
4./3
(3/3)^ -SrFrac
3 - ? / ? 4 - S F F r a27/E
s
I
(?\'
\ J./
\3/
1
222
| _27
88
27
JJJ
.r.-.1
N o t et h a t ( i )
-
,r.3
(;)
1
c..
o1,'
?/9
-
123
b-
I
-
-1
Notethat4:4
36
6'
2r
* FigureP.13
These three expressionsare shown on a graphing calculator in Fig. P.13.Note
ffi
that the fractional base must be in parentheses.The fraction feature was used to set
fractional answers.
\rV \I,rt. Simplif,.a.2-2. 43 a.
. tz-'
O-'
Example 1(b) illustrates the fact that a fractional basecan be inverted, ifthe sign
of the exponent is changed.Example I (c) illustrates the fact that a factor of the numerator or denominator can be moved from the numerator to the denominator or
vice versa as long as we change the sign ofthe exponent. These rules follow from
the definition of negative exponents.
Rulesfor Negative
Exponentsand Fractions
Ifa and b arenonzeroreal numbers andmandn are integers,then
#:#
e)-^:H'and
Using this rule, we could shortenExamples1(b) and (c) as follows:
+
ac
)/3^ -SIFF
SE/E
( 1+?*S)/3*?rFras
/z\-3
/:\3
27
6-22382
2-t
62
36
9'
Note thatwe cannotapplytheseruleswhenadditionor subtractionis involved.
I
* FigureP.14
and
t;i:(.t:T
l+3-2
+' 1 + 2 3
2-3
32
ffi
FigureP.14showstheseexpressions
on a graphingcalculator. tr
Rutesof Exponents
Considerthe product a2 . a3. Using the definition of exponents,we can simplify
this product as follows:
a2. a3 :
(a. a)(a. a. a) : os
Similarly, if m and n are any positive integerswe have
m factors
a-.an:a.a
ri factors
a.a.a.....a:at*n
m -f n factors
This equation indicates that the product of exponential expressionswith the same
base is obtained by adding the exponents.This fact is called the product rule.
P.2r. €
fuanFla I
IntegralExponents
and Scientific
Notation
17
Usingthe productrule
Simplifueachexpression.
a. (3x9yz)(-zxy47 b. 23 - 32
Sotution
a. Usetheproductrule to addthe exponents
whenbasesareidentical:
(zx8y2)(-zxyo): - 6*'yu
b. Sincethe basesaredifferentwe cannotusethe productrule, but we cansimplifz
theexpression
usingthedefinitionofexponents:
23'32 :8'9
:72
4rV 7fu1. Simplify-2ct4b3(-3a5b6).
So far we have defined positive and negative integral exponents.The definition
of zero as an exponentis given in the following box. Note that the zero power of zero
is not defined.
Definition:ZeroExponent
+
-3+3
If a is a nonzero real number, then a0 :
l.
The definition of zero exponent allows us to extend the product rule to any
integral exponents.For example, using the definition of negative exponents,we
set
t
3. 125
. FigureP.15
Rulesfor
IntegralExponents
2-3 .
^l
1)
23: l.
Adding exponents,we get 2-3 . 23 : 2-3+3 : 20.The answeris the samebecause
2u is defined to be 1.
To evaluate2-3+3 on a calculator,the expression-3 * 3 must be in parenffi
thesesas in Fig. P.15. tr
Using the definitions of positive, negative,and zero exponents,we can show
that the product rule and several other rules hold for any integral exponents.We list
theserules in the followins box.
If a and 6 are nonzero real numbers andm andn are integers,then
l. a*en : sm*n
Product rule
'' # * am-n
Quotientrule
3. {an)n : qmn
4. (ab):
Q'bo
Powerof a power rule
Powerof a prodirct rule
:#
'.G)"
Powerof a quotient rule
The rules for integral exponentsare used to simplify expressions.
18
ChapterP rrt
Prerequisites
foam/e
with
Simptifyingexpressions
integralexponents
I
Assume
Simplifr eachexpression.
Write your answerwithout negativeexponents.
that all variablesrepresentnonzerorealnumbers.
2y-s),.
a. (3x2y3)(-4x
#S
Sotution
a. (3x2y3)(-4x-2y-\:
-12x2+(-z)y3+(-5)
Product
rule
-12*oy-z
Simplifu the exponents.
t2
*-v
Definition ofnegative and zero exponents
"v
-6a'b'
D . ^ ? - 1zao
:
- J^a. --
:
-3o-262
7b-l-e3\
Quotient rule
Simplify the exponents.
3b2
,)
Definition of negative exponents.
u
77V Tht.
Simplify-to-t6-s(9a-2b8).
that
In the next example,we usethe rulesof exponentsto simpli$ expressions
havevariablesin the exponents.
fua*/e
with
Simptifyingexpressions
variabteexponents
I
Simplify each expression.Assume lhat alI bases are nonzero real numbers and all
exponentsare integers.
a. (-3xa-sr-s1+ r.
/z^2n-t\-z
\;;*)
Sotution
a. (-3xo-sy-')o :
(-21+1ro-s1+j-t)o
power
of a product
rute
:8lx4a-2oy-12
Powerofapowerrule
: Yroo-'o
Definition of negative exponents
;E-
.
(3a2,-r1-s
u'l^-Ln1-
\za
--/
3-t1ozm-t1-t
^-1r-1-,-1
z -\a -') -
23o-6n+3
33o9m
Power of a quotient rule
Power of a power rule and definition
ofnegative exponents
gO-tsn+3
27
Quotient rule
1ry Tbl. simplify(2a'-z1z(-2oo*)3.