Middle School Math & Grammar/ProOneSoftware-MiddleSchoolMath&Grammar-Win31.iso/math/chapter1.6b

Text File | 1994-11-22 | 5.6 KB | 348 lines

 346
àï1.6ïExponents and Order of Operations.
äïPlease write the following products of Whole Numbers
êêusing exponents.
âS

#êêï1)è 2 ∙ 2 ∙ 2è=è2Ä


#êêï2)è 3 ∙ 3 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4è=è3ì ∙ 4É
éS In order to write the product, 3 ∙ 3 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4,
using exponents, it is necessary to count the number of times 3 repeats
as a factor and the number of times 4 repeats as a factor.ïSince the 3
occurs twice and the 4 occurs five times, the product can be written
#asï3ì ∙ 4É.ïIn this expression "3" is called a "base" and "2" is
called an exponent.ïSimilarly 4 is a base and 5 an exponent.
#êêï3 ∙ 3 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4è=è3ì ∙ 4É
 1
êëWrite the product, 5 ∙ 5 ∙ 5, using an exponent.


#ë A)ï15Äêè B)ï3ÉêëC)ï5ÄêëD)ïå
ü


#êêêë5 ∙ 5 ∙ 5è=è5Ä
Ç C
 2
êè Write the product, 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 , using an exponent.


#ë A)ï4æêëB)ï6 ∙ 4êïC)ï16Åêè D)ïå
ü


#êêê4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4è=è4æ
Ç A
 3
êWrite the product, 3 ∙ 3 ∙ 3 ∙ 5 ∙ 5 ∙ 5 ∙ 5, using exponents.


#ë A)ï27 ∙ 4ÉêB)ï3Ä ∙ 5ÅêC)ï3Å ∙5Äê D)ïå
ü


#êê 3 ∙ 3 ∙ 3 ∙ 5 ∙ 5 ∙ 5 ∙ 5è=è3Ä ∙ 5Å
Ç B
 4
êWrite the product, 2 ∙ 2 ∙ 5 ∙ 5 ∙ 5 ∙ 6 ∙ 6 ∙ 6 ∙ 6, using
êexponents.

#èA)ï10Ä ∙ 6ÅêB)ï2ì ∙ 5Ä ∙ 6Åë C)ï(2 ∙ 5 ∙ 6) öè D) å
ü


#êë2 ∙ 2 ∙ 5 ∙ 5 ∙ 5 ∙ 6 ∙ 6 ∙ 6 ∙ 6è=è2ì ∙ 5Ä ∙ 6Å
Ç B
 5
ë Write the product, 6 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10, using
ë exponents.

#èA)ï6îò ∙ 7ê B)ï6 ∙ 7 ∙ 10ë C)ï6 ∙ 10ÆêïD) å
ü


#êè6 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10è=è6 ∙ 10Æ
Ç C
äïPlease simplify the following exponential expressions.
âS
#êêêï4Äè=è4 ∙ 4 ∙ 4è=è64


#êêï2Ä ∙ 5ìè=è2 ∙ 2 ∙ 2 ∙ 5 ∙ 5è=è200
éS

#ë In order to simplify the expression, 2Ä ∙ 5ì, it is
necessary to write the "2" as a factor three times and the "5" as a
factor twice.
#êêë2Ä ∙ 5ìè=è2 ∙ 2 ∙ 2 ∙ 5 ∙ 5

Next, you should multiply the factors in the product, 2∙2∙2∙5∙5, and
get the answer 200.
#êêï2Ä ∙ 5ìè=è2 ∙ 2 ∙ 2 ∙ 5 ∙ 5è=è200
 6
#êêêêSimplifyè 3Å


êA)ï81êëB)ï12êè C)ï24êëD)ïå
ü


#êêê3Åè=è3 ∙ 3 ∙ 3 ∙ 3è=è81
Ç A
 7
#êêêëSimplifyè 2Ä ∙ 4ì.


êA)ï120êè B)ï64êè C)ï128êè D)ïå
ü


#êêè2Ä ∙ 4ìè=è2 ∙ 2 ∙ 2 ∙ 4 ∙4è=è128
Ç C
 8
#êêêëSimplifyè 3Å ∙ 5ì.


êA)ï250êè B)ï2,025ê C)ï120êè D)ïå
ü


#êë 3Å ∙ 5ìè=è3 ∙ 3 ∙ 3 ∙ 3 ∙ 5 ∙ 5è=è2,025
Ç B
 9
#êêêèSimplifyè 2Å ∙ 3ì ∙ 10.


êA)ï480êè B)ï1,440ê C)ï236êè D)ïå
ü


#ê 2Å ∙ 3ì ∙ 10è=è2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ∙ 3 ∙ 10è=è1,440
Ç B
 10
#êêêèSimplifyè 7 ∙ 2Ä ∙ 10ì.


êA)ï5,600êïB)ï4,200ê C)ï286êè D)ïå
ü


#êï7 ∙ 2Ä ∙ 10ìè=è7 ∙ 2 ∙ 2 ∙ 2 ∙ 10 ∙ 10è=è5,600
Ç A
 11
#êêêïSimplifyè 5Ä ∙ 2ì ∙ 10Å.


êA)ï62,426ê B)ï4,280ê C)ï5,000,000ëD)ïå
ü


#5Ä ∙ 2ì ∙ 10Åè=è5 ∙ 5 ∙ 5 ∙ 2 ∙ 2 ∙ 10 ∙ 10 ∙ 10 ∙ 10ï=ï5,000,000
Ç C
 12
#êêêïSimplifyè 2Ä ∙ 3Å ∙ 4É.


êA)ï663,552êB)ï4,622ê C)ï384êè D)ïå
ü


#2Ä ∙ 3Å ∙ 4Éï=ï2 ∙ 2 ∙ 2 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 4 ∙ 4 ∙ 4 ∙4 ∙4ï=ï745,472
Ç A
ä Please simplify the following expressions using the
êë correct Order of Operations.
#âêê 4 ÷ 2 + (6 - 5) + 5 ∙ 3 - 2ì
êêê 4 ÷ 2 + (6 - 5) + 5 ∙ 3 - 4
êêê 4 ÷ 2 +è 1è + 5 ∙ 3 - 4
êêêè2è+è 1è + 5 ∙ 3 - 4
êêêè2è+è 1è +ï15è- 4
êêêê3êï+ï15è- 4
êêêêê 18ê - 4è=è14
éS
#è To simplify the expression, 4 ÷ 2 + (6 - 5) + 5 ∙ 3 - 2ì, it
is necessary to perform operations inside of parençs first.
#êêêè 4 ÷ 2 + 1 + 5 ∙ 3 - 2ì

The next thing to do is simplify exponents.
êêêè 4 ÷ 2 + 1 + 5 ∙ 3 - 4

Next, you should perform all of the multiplication and division as they
occur from left to right.
êêêê2 + 1 + 5 ∙ 3 - 4
êêêê2 + 1 + 15 - 4

Finally, you should perform all of the addition and subtraction as they
occur from left to right.
êêêêï3 + 15 - 4
êêêêï18 - 4
êêêêï14

Thus, the value of the expression is 14.ïIf you simplify this
expression using the wrong Order of Operations, you will usually get a
different answer.ïIt is necessary to do things in the following order:
ê 1)ïSimplify operations in parençs.
ê 2)ïSimplify exponents.
ê 3)ïPerform multiplication and division operations first come
êëfirst serve moving from left to right.
ê 4)ïPerform addition and subtraction operations first come first
êëserve moving from left to right.
 13
êêêèSimplifyë 4 ÷ 2 ∙ 3

êï2
#ëA)ï─êë B)ï6êëC)ï12êë D)ïå
êï3
ü
êêêêè4 ÷ 2 ∙ 3

êêêêë2 ∙ 3

êêêêê6
Ç B
 14
êêêèSimplifyë 2 ∙ 12 ÷ 3.


ëA)ï8êë B)ï12êè C)ï18êë D)ïå

ü
êêêêè2 ∙ 12 ÷ 3

êêêêë24 ÷ 3

êêêêê 8
Ç A
 15
êêêè Simplifyë 4 + 2 - 3


ëA)ï9êë B)ï5êëC)ï3êêD)ïå

ü
êêêêè4 + 2 - 3

êêêêë6 - 3

êêêêê3
Ç C
 16
êêêè Simplifyë 12 ÷ 4 + 2.


ëA)ï5êë B)ï2êëC)ï14êë D)ïå

ü
êêêêï12 ÷ 4 + 2

êêêêë3 + 2

êêêêê5
Ç A
 17
êêêèSimplifyë 4 ∙ 3 + 5


ëA)ï14êëB)ï32êè C)ï17êë D)ïå

ü
êêêêè4 ∙ 3 + 5

êêêêë12 + 5

êêêêê17
Ç C
 18
#êêêè Simplifyë 4ì - 8.


ëA)ï0êë B)ï8êëC)ï12êë D)ïå

ü
#êêêêè 4ì - 8

êêêêè 16 - 8

êêêêê8
Ç B
 19
êêêSimplifyë 3 + (4 + 2) ÷ 3.


ëA)ï12êëB)ï5êëC)ï8êêD)ïå

ü
êêêê 3 + (4 + 2) ÷ 3

êêêêè3 + 6 ÷ 3
êêêêë3 + 2
êêêêê5
Ç B
 20
êêë Simplifyë 12 ∙ (4 - 2) ÷ 8


ëA)ï12êëB)ï24êè C)ï3êêD)ïå

ü
êêêë 12 ∙ (4 - 2) ÷ 8

êêêêï12 ∙ 2 ÷ 8
êêêêè 24 ÷ 8
êêêêê3
Ç C
 21
#êêêSimplifyë 2Ä + 6 ∙ (9 - 7).


ëA)ï6êë B)ï20êè C)ï12êë D)ïå

ü
#êêêê2Ä + 6 ∙ (9 - 7)
êêêê 8 + 6 ∙ (9 - 7)
êêêêè 8 + 6 ∙ 2
êêêêë8 + 12
êêêêê20
Ç B
 22
êêë Simplifyë 8 + 4 - 2 ∙ 5 ÷ 2.


ëA)ï7êë B)ï16êè C)ï24êë D)ïå

ü
êêêë8 + 4 - 2 ∙ 5 ÷ 2
êêêê8 + 4 - 10 ÷ 2
êêêêï8 + 4 - 5
êêêêè 12 - 5
êêêêê7
Ç A
 23
êêêSimplifyë 10 - (8 - 4) ÷ 2.


ëA)ï8êë B)ï6êëC)ï4êêD)ïå

ü
êêêê10 - (8 - 4) ÷ 2
êêêêè10 - 4 ÷ 2
êêêêë10 - 2
êêêêê 8
Ç A
 24
#êêëSimplifyë 5 ∙ (9 - 3) - 2ì.


ëA)ï24êëB)ï18êè C)ï26êë D)ïå

ü
#êêêë 5 ∙ (9 - 3) - 2ì
êêêê5 ∙ (9 - 3) - 4
êêêêï5 ∙ 6 - 4
êêêêè 30 - 4
êêêêê26
Ç C