[PDF] Integral Exponents and Scientific Notation

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944_6Math110cPs2a.pdf c) d) P.2 .s s Integral Exponents and Scientific Notation 15 Sandy has one quart ofgrass seed and one quart ofsand, each stored in one gal- lon containers. Sandy pours a little seed into the sand and shakes well. She then pours the same amount of the mix back into the container of seed so that both containers again contain exactly one quart. Is there more sand in Sandy's seed or more seed in Sandy's sand? Explain. The percentage ofseed ir.r the sand is equal to the percentage ofsarrd in the seed. In each cell of the second row 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 second row must indi-

cate the number of times that the digit above it appears in the second row.6.2. t,0.0.0.1,0.0,0 i*0 *-i-f-l;- [-3--T;"T;*ru-I ? -1 ]*i;-] i--**-l^-"*li-**- {- - -*i* -"} * -:.+ '**-T-**1-***l-**- I We defined positive integral exponents in Section P.1. In this section we will define negative integral exponents and review the rules for exponents. Then we will see how exponents are used in scientific notation to indicate very large and very small numbers.

Negative Integral Exponents

We use a negative sign in an exponent to represent multiplicative inverses or recipro- cals. For negative exponents we do not allow the base to be zero because zero does not have a reciprocal. i*-*-**-*'*"*..1: : If o i, a nonzero real number and n is a positive integer, th"n o-n : ). I

Integral Exponents and Scientific Notation

Definition: Negative

Integral Exponents

foa.nF/a I rvatuating expressions that have negative exponents Simplify each expression without using a calculator, then check with a calculator. a. 3-r . s-z .to2 o. /?\-' ^ 6-2 \l/ c' 24

Solution

a.3r.5r.lo2llt00 : -.-. 100325 100 4
753
lt J)-

16Chapter P I sffiPrerequisites

4./3(3/3)^ -SrFrac

27/E3 -?/?4 -SFFras

?/9 -1+5* -2+1Br FFF * Figure P.13 o(3)' \ J./ 1- b- I ( ?\' \3/ | _27 88
1 222
36
-16' .r.-.1 ,r.3Note that (i) - (;) JJJ27 o-c.. 1,'

123Notethat4:4I-2r

ffi These three expressions are shown on a graphing calculator in Fig. P.13. Note that the fractional base must be in parentheses. The fraction feature was used to set

Rules for Negative

Exponents and Fractions

* Figure P.14 fractional answers. \rV \I,rt. Simplif,. a.2-2 . 43 a. O-' . tz-' /z\-3 /:\3 27t;i:(.t:T and Ifa and b arenonzeroreal numbers andmandn are integers, then e)-^:H'and #:# Example 1(b) illustrates the fact that a fractional base can be inverted, ifthe sign of the exponent is changed. Example I (c) illustrates the fact that a factor of the nu- merator 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. Using this rule, we could shorten Examples 1(b) and (c) as follows:

6-22382

2-t 62 36 9'

Note that we cannot apply these rules when addition or subtraction is involved. l+3-2 +1+232-3 ' 32 ffi Figure P.14 shows these expressions on a graphing calculator. tr

Rutes of Exponents

Consider the 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 integers we have m factors ri factors a-.an:a.a a.a.a.....a:at*n m -f n factors This equation indicates that the product of exponential expressions with the same base is obtained by adding the exponents. This fact is called the product rule. +)/3^ -SIFF acSE/E( 1+?*S)/3*?rFras I

Definition: Zero Exponent

P.2 r. € Integral Exponents and Scientific Notation17 fuanFla I Using the product rule

Simplifu each expression.

a. (3x9yz)(-zxy47 b. 23 - 32

Sotution

a. Use the product rule to add the exponents when bases are identical: (zx8y2)(-zxyo) : - 6*'yu b. Since the bases are different we cannot use the product rule, but we can simplifz the expression using the definition ofexponents:

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 exponent is given in the following box. Note that the zero power of zero is not defined.

If a is a nonzero real number, then a0 : l.

+t-3+3

3. 125

. Figure P.15 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

2-3 .23: l.

Adding exponents, we get 2-3 . 23 : 2-3+3 : 20. The answer is the same because

2u is defined to be 1.

ffi To evaluate 2-3+3 on a calculator, the expression -3 * 3 must be in paren- theses as 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 these rules in the followins box. ^l 1)

Rules for

Integral Exponents

If a and 6 are nonzero real numbers andm andn are integers, then l. a*en : sm*n Product rule '' # * am-n

3. {an)n : qmn

4. (ab): Q'bo'. G)" :#

Quotient rule

Power of a power rule

Power of a prodirct rule

Power of a quotient rule

The rules for integral exponents are used to simplify expressions.

18ChapterP rrt Prerequisites

foam/e I Simptifying expressions withintegral exponents Simplifr each expression. Write your answer without negative exponents. Assume that all variables represent nonzero real numbers. a. (3x2y3)(-4x 2y-s) ,. #S

Sotution

a. (3x2y3)(-4x-2y-\: : -3o-262 3b2,) u -12x2+(-z)y3+(-5) Product rule -12*oy-z t2*-v "v

Simplifu the exponents.

Definition ofnegative and zero exponents

Quotient rule

Simplify the exponents.

Definition of negative exponents.

- -6a'b' ^.-D. ^ ?-1 : -Ja- 7b-l-e3\ zao -

Power of a quotient rule

Power of a power rule and definition

ofnegative exponents

Quotient rule

77V Tht. Simplify -to-t6-s(9a-2b8).

In the next example, we use the rules of exponents to simpli$ expressions that have variables in the exponents. fua*/e I Simptifying expressions with variabte exponents Simplify each expression. Assume lhat alI bases are nonzero real numbers and all exponents are integers. /z^2n-t\-za. (-3xa-sr-s1+ r. \;;*)

Sotution

a. (-3xo-sy-')o : (-21+1ro-s1+j-t)o power of a product rute :8lx4a-2oy-12 Powerofapowerrule : Yroo-'o ;E-

Definition of negative exponents

. (3a2,-r1-s 3-t1ozm-t1-t u'l^-Ln1- ^-1r-1-,-1\za --/ z -\a -') -

23o-6n+3

33o9m
gO-tsn+3 27

1ry Tbl. simplify (2a'-z1z (-2oo*)3.

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