Content Standard
CC-8
CC-8
Using Rational
Exponents
Objective
N.RN.1 Explain how the definition of the meaning of
rational exponents follows from extending the properties
of integer exponents . . . .
Solve It!
PURPOSE 5PQSFTFOUTUVEFOUTXJUIBWBSJFUZPG
FYQSFTTJPOTUIBUJODMVEFJOUFHFSFYQPOFOUT
PROCESS 4UVEFOUTNBZ
t VTFUIFSVMFTPGFYQPOFOUTUPTJNQMJGZFBDI
FYQSFTTJPOUPBTJOHMFCBTFBOEFYQPOFOU
t DPNQBSFBOEPSEFSTJNQMJGJFEFYQSFTTJPOT
To extend operations with powers to rational exponents
22 t 2
" !
!
(2–1 )2
(22)2 t 2–2
2–1
2t
2
t
FACILITATE
Q 8IBUJTBOPUIFSXBZUPXSJUFBTBQPXFSXJUIBO
2
2–2
FYQPOFOU [21]
Q &YQMBJOUIFEJGGFSFODFCFUXFFOUIFFYQSFTTJPOT
1 BOE (1) [To simplify the first
expression, add the exponents. To simplify the
second expression, multiply the exponents.]
(22)2
MATHEMATICAL
PRACTICES
1 Interactive Learning
Not all exponents are integers. Exponents may be rational numbers, such as fractions.
Essential Understanding You can use properties of exponents to simplify
expressions with rational exponents.
Q 8IJDIJTHSFBUFSPS 1 &YQMBJO [2 is
greater because 21 2 20 , or 1.]
Property
Multiplying Powers With the Same Base
ANSWER 4FF4PMWF*UJO"OTXFSTPOOFYUQBHF
CONNECT THE MATH *OUIJTMFTTPOTUVEFOUTXJMM
Words To multiply powers with rational exponents and the same base, add the exponents.
Algebra am an amn , where a 0 and m and n are rational numbers
1
1
1
1
2
Examples 2 3 2 3 2 3 3 2 3
Problem 1
1
2
1
2
MFBSOUIBUUIFQSPQFSUJFTPGJOUFHFSFYQPOFOUTBMTP
BQQMZUPSBUJPOBMFYQPOFOUT4UVEFOUTNVTUCF
QSPGJDJFOUXJUIDPNQVUJOHXJUIJOUFHFSFYQPOFOUT
BTXFMMBTQFSGPSNJOHPQFSBUJPOTPOGSBDUJPOT
4UVEFOUTXJMMDPOOFDUSBUJPOBMFYQPOFOUTUPSPPUTJO
"MHFCSB
3
d5 d5 d55 d5
Multiplying Powers With Rational Exponents
What is the simplified form of each expression?
3
1
3
1
A 510 510 510 10
4
What extra step
is needed with
fractional exponents?
The denominators must
be the same to perform
addition.
Add the exponents of powers with the same base.
2
Simplify the exponent.
5 10 5 5
B
1
2
1
2
Add the exponents of powers with the same base.
1
6
Write the fractions with like denominators.
c9 c3 c93
c99
7
Take Note
Simplify.
c9
Got It?
2 Guided Instruction
1. What is the simplified form of each expression?
3
5
1
a. 15 7 15 7
3
b. z 5 z 10
v
c. 3 v 3 7
CC-8 Using Rational Exponents
1
Problem 1
CC-8 Preparing to Teach
BIG ideas Properties
Equivalence
ESSENTIAL UNDERSTANDINGS
t&YQSFTTJPOTXJUISBUJPOBMFYQPOFOUT
DBOCFTJNQMJGJFEVTJOHQSPQFSUJFTPG
FYQPOFOUT
Math Background
*OUIJTMFTTPOTUVEFOUTDPOOFDUXIBU
UIFZIBWFQSFWJPVTMZMFBSOFEBCPVUUIF
QSPQFSUJFTPGJOUFHFSFYQPOFOUTUPUIF
QSPQFSUJFTPGSBUJPOBMFYQPOFOUT4UVEFOUT
TIPVMECFBCMFUPNVMUJQMZQPXFSTXJUI
UIFTBNFCBTFSBJTFBQPXFSUPBQPXFS
BOESBJTFBQSPEVDUUPBQPXFS4UVEFOUT
TIPVMEBMTPCFGBNJMJBSXJUIBEEJOHBOE
NVMUJQMZJOHGSBDUJPOT4UVEFOUTXJMMBQQMZ
UIJTLOPXMFEHFUPEJTDPWFSBOEVTFUIF
QSPQFSUJFTPGSBUJPOBMQPXFST
3FWJFXXJUITUVEFOUTUIFDPNQPOFOUTPGB
QPXFSJODMVEJOHCBTFBOEFYQPOFOU5IFSVMFGPS
NVMUJQMZJOHQPXFSTXJUIUIFTBNFCBTFSFRVJSFT
UIBUUIFCBTFPGFBDIGBDUPSCFJEFOUJDBM5IFCBTF
NBZCFBOVNCFSPSBWBSJBCMF$PNQBSFUIFSVMF
GPSNVMUJQMZJOHQPXFSTXJUIUIFTBNFCBTFBOE
JOUFHFSFYQPOFOUTXJUINVMUJQMZJOHQPXFSTXJUIUIF
TBNFCBTFBOESBUJPOBMFYQPOFOUT
Q 8IZJTBOFYUSBTUFQOFFEFEUPTPMWF# [The
Mathematical Practice
Look for and make use of structure.
*OEJTDFSOJOHUIFQSPQFSUJFTPGSBUJPOBM
FYQPOFOUTTUVEFOUTXJMMDPOOFDUUIF
QSPQFSUJFTPGJOUFHFSFYQPOFOUTUPSBUJPOBM
FYQPOFOUT5IFZXJMMBQQMZUIFTUSVDUVSF
PGJOUFHFSFYQPOFOUTUPUIFTUSVDUVSFPG
SBUJPOBMFYQPOFOUT
denominators are different, so you
need to rewrite the fractions with like
denominators before adding.]
Got It?
Q 8IBUJTUIFQSPDFTTGPSNVMUJQMZJOHQPXFSTXJUI
UIFTBNFCBTFJGPOFPSNPSFPGUIFFYQPOFOUTBSF
OFHBUJWF [Add the exponents, accounting
for negative signs, as you would when
adding negative fractions.]
CC-8
1
Take Note
Property Raising a Power to a Power
$PNQBSFUIFSVMFGPSSBJTJOHBQPXFSUPBQPXFS
XJUIJOUFHFSFYQPOFOUTUPUIFSVMFGPSSBJTJOHB
QPXFSUPBQPXFSXJUISBUJPOBMFYQPOFOUT'PS
JOUFHFSTBOESBUJPOBMFYQPOFOUTNVMUJQMZUIF
FYQPOFOUT
Words To raise a power with a rational exponent to a power, multiply the exponents.
Algebra (am)n amn , where a 0 and m and n are rational numbers
1 3
1 3
1 3
3
Examples 4 2 5 4 2 5 4 10
Problem 2
1 3
Problem 2 Simplifying Powers of Powers
Q )PXEPFTSBJTJOHBQPXFSXJUIBSBUJPOBMFYQPOFOU
UPBQPXFSEJGGFSGSPNNVMUJQMZJOHQPXFSTXJUI
SBUJPOBMFYQPOFOUTBOEUIFTBNFCBTF [To raise
a power to a power, multiply the exponents. To
multiply two powers, add the exponents.]
How do you raise a
power to a power
when the exponents
are rational?
The process is the same
as for integer exponents:
multiply the exponents.
What is the simplified form of each expression?
1 2
1 2
A 64 7 64 7
Multiply exponents when raising a power to a power.
1
14
Simplify.
6
B
3 1
3 1
Multiply exponents when raising a power to a power.
3
Simplify.
k 8 2 k 8 2
k 16
Got It?
Got It? 2. What is the simplified form of each expression?
Q 8IBUSVMFT
DPODFSOJOHBEEJUJPOXJUIQPTJUJWF
BOEOFHBUJWFJOUFHFSTBQQMZUPBEEJUJPOXJUI
QPTJUJWFBOEOFHBUJWFSBUJPOBMFYQPOFOUT
2
3
3
3
3
the whole-number base as a power. For example, 164 (2 2 2 2)4 (24)4 24 4 23 8.
Property Raising a Product to a Power
Words To raise a product to a rational power, raise each factor to the power and multiply.
Algebra (ab)n anbn , where a 0, b 0, and n is a rational number
2
2
2
2
2
2
2
2
2
Examples (27u)3 273u3 (3 3 3)3u3 (33)3u3 32u3 9u3
[The rules are the same for integers and
rational exponents: positive times positive
is positive; negative times negative
is negative; positive times negative is
negative.]
Problem 3 Simplifying Powers of Products
What is the simplified form of each expression?
1
A (32m)5
Raise each factor to the
one-fifth power.
ERROR PREVENTION
Simplify the expression.
2
Common Core
Answers
Additional Problems
Solve It!
(1) 1 () ()
b. z
1. 8IBUJTUIFTJNQMJGJFEGPSNPGFBDI
FYQSFTTJPO
1
1
a. 78 78
1
4
b. k 5 k 15
1
Got It?
ANSWER a7 4
1. a.15
7
7
15
b. k
2. 8IBUJTUIFTJNQMJGJFEGPSNPG
FBDIFYQSFTTJPO
1
8v
c. 3 7
2. a. 10 3 PS
1
b. a 1
3
1
1
b. (t 3 6
a. 54 1
103
3
1
ANSWER a 5 8 PS 3 58
1
9
b. t
3. 8IBUJTUIFTJNQMJGJFEGPSNPGFBDI
FYQSFTTJPO
1
1
b. 81z 3 4
a. 64c 3
1
ANSWER a. 4c 3
3
b. 3z 4
CC-8
4USFTTUIFJNQPSUBODFPGLOPXJOHXIFOUPBEEUIF
FYQPOFOUTBOEXIFOUPNVMUJQMZ4PNFTUVEFOUT
NBZNVMUJQMZUIFFYQPOFOUTXIFONVMUJQMZJOH
FYQSFTTJPOTXJUISBUJPOBMFYQPOFOUTBOEUIFTBNF
CBTFSBUIFSUIBOBEEJOHUIFFYQPOFOFUT
3FWJFXXJUITUVEFOUTUIFDPNQPOFOUTPGB
QSPEVDUUPBQPXFSJODMVEJOHBCBTFDPNQSJTFE
PGTFWFSBMGBDUPSTBOEBOFYQPOFOU5IFQSPEVDU
abJTDPNQSJTFEPGUIFUXPGBDUPSTaBOEb5IF
QSPEVDUuJTDPNQSJTFEPGUIFUXPGBDUPST
BOEu&BDIGBDUPSNVTUCFSBJTFEUPUIFSBUJPOBM
QPXFSJOEJWJEVBMMZ/PUFUIBUTPNFGBDUPSTDBOCF
TJNQMJGJFEGVSUIFSCZSFXSJUJOHUIFGBDUPSBTBCBTF
BOEFYQPOFOU3
1 3
b. a9 4
You may be able to simplify a power with a whole-number base and a rational exponent by rewriting
2 8IBUSVMFT
DPODFSOJOHNVMUJQMJDBUJPOXJUIQPTJUJWF
BOEOFHBUJWFJOUFHFSTBQQMZUPNVMUJQMJDBUJPO
XJUIQPTJUJWFBOEOFHBUJWFSBUJPOBMFYQPOFOUT Take Note
5
a. 105 3
[The rules are the same for integers and
rational exponents: positive plus positive
is positive; negative plus negative is
negative; positive plus negative may be
positive or negative.]
2
1
r 3 4 r 3 4 r 4
1
B (3f 8) 4
How are the two
parts of Problem 3
different?
When the variable is
raised to a power, you
need to use the powerof-a-power property to
simplify the variable part
of the expression.
Problem 3
Raise each factor to the
one-fourth power.
Q 'PS"XIBUBSFUIFGBDUPSTPGUIFCBTF [32 and
m]
Multiply the exponents
of a power raised to a
power.
Simplify the expression.
Q )PXDBOZPVSFXSJUFBTBQPXFSXJUIBCBTF
BOEBOFYQPOFOU [25]
Q 'PS#XIBUBSFUIFGBDUPSTPGUIFCBTF [3 and
ƒ 8]
Got It?
Got It? 3. What is the simplified form of each expression?
b. (9w2)2
TUVEFOUTNBZNVMUJQMZUIFGBDUPSTCZ 14 SBUIFSUIBO
Lesson Check
Do you know HOW?
Do you UNDERSTAND?
1
10
5
12
1. 36 36
2. v 7
v10
6
9 1
7 9
4. (h6)5
1
6. (4s4)2
5
3. 10 1
5. (8j)3
3 Lesson Check
Practice and Problem-Solving Exercises
A
GJOEJOHUIF 14 UIQPXFS"MTPTPNFTUVEFOUTNBZ
POMZSBJTFPOFPGUIFGBDUPSTPGUIFCBTFUPUIF
FYQPOFOU4USFTTUIBUJOPSEFSUPTJNQMJGZFBDI
GBDUPSOFFETUPCFSBJTFEUPUIFFYQPOFOU
MATHEMATICAL
PRACTICES
7. Reasoning Explain how multiplying expressions
with the same base and integer exponents is similar
to multiplying expressions with the same base and
rational exponents.
Simplify each expression.
1
12
Practice
Do you know HOW?
MATHEMATICAL
PRACTICES
1
1
9. 24 24
1
1
2
1
10. 83 83
1
1
11. 510 510
2
2
13. y 8 y 8
7
3
1
1
14. b2 b2
15. m12 m12
8. 96 96
12. x 5 x 5
1
1
See Problem 2.
Simplify each expression.
3 2
4 5
1 1
8 8
16. 6 18. 9 2
4 7
7 8
20. p 2 1
3 2
1
3
17. 2 2 5
9 9
21. h 22. c
1
10
19. 27 9
5
23. z25 2
6
10
1
1
1
1
26. 3n2 8
25. 27x 3
27. 64x9 3
3
CC-8 Using Rational Exponents
Answers
Practice and Problem-Solving
Exercises
Got It? (continued)
8. 9 3
1
10. 83 PS 1
1
9. 1
11. 55 PS 11
4
13. y 4
1
15.m 6 PS 1
m
1
17. 64
19.
1
21. h 3
23. z 5
1
4
3. a. b
b. 3w
Lesson Check
2. v
3. 10
1
7
1
Do you UNDERSTAND?
See Problem 3.
Simplify each expression.
24. 36k 2
t*GTUVEFOUTIBWFEJGGJDVMUZLOPXJOHXIFOUPBEE
BOEXIFOUPNVMUJQMZSBUJPOBMFYQPOFOUTUIFO
IBWFUIFNMPPLBU1SPCMFNTBOE3FWJTJUUIF
TUSVDUVSFPGBQPXFSXIJDIDPOTJTUTPGBCBTF
BOEBOFYQPOFOU4UVEFOUTTIPVMECFBCMFUP
EJGGFSFOUJBUFCFUXFFOBOFYQSFTTJPOJOXIJDIUXP
QPXFSTBSFNVMUJQMJFEBOEBOFYQSFTTJPOJOXIJDI
BQPXFSJTSBJTFEUPBQPXFS.BLFTVSFUIFZDBO
BQQMZUIFBQQSPQSJBUFSVMFTUPTJNQMJGZ
See Problem 1.
Simplify each expression.
1.
ERROR PREVENTION
4UVEFOUTXJMMOPUDPOOFDUSBUJPOBMFYQPOFOUTUP
SPPUTVOUJM"MHFCSB"MMPXTUVEFOUTUPVTFB
DBMDVMBUPSUPTJNQMJGZXIFOQPTTJCMF'PSBTPNF
1
1
a. (16b)4
53
PS 1
3
v5
4. h 1 PS 1h
5. j 3
6.T
7. 8IFOZPVNVMUJQMZQPXFSTXJUI
JOUFHFSFYQPOFOUTBOEUIFTBNFCBTF
BEEUIFFYQPOFOUT4JODFUIFSVMFT
GPSBEEJOHJOUFHFSTBSFUIFTBNFBT
UIPTFGPSBEEJOHSBUJPOBMOVNCFST
ZPVBMTPNVMUJQMZQPXFSTXJUISBUJPOBM
FYQPOFOUTBOEUIFTBNFCBTFCZ
BEEJOHUIFFYQPOFOUT
12. x 5
14.
3
16. 6 10
18. 9 3
10
20. p 81
9
22. c 100 PS
24. 6k
1
1
1
26. 3 8n 4
1
1
9
c 100
Close
Q *OXIBUUZQFPGFYQSFTTJPOTIPVMEZPVBEEUIF
SBUJPOBMFYQPOFOUTUPTJNQMJGZBOEJOXIBU
UZQFPGFYQSFTTJPOTIPVMEZPVNVMUJQMZUIF
SBUJPOBMFYQPOFOUTUPTJNQMJGZ [Add rational
55
5
t*GTUVEFOUTIBWFEJGGJDVMUZDPOOFDUJOHUIFSVMFTGPS
NVMUJQMZJOHXJUIJOUFHFSFYQPOFOUTUPUIFSVMFT
GPSNVMUJQMZJOHXJUISBUJPOBMFYQPOFOUTUIFO
IBWFUIFNMPPLBU1SPCMFN*GTUVEFOUTBSFTUJMM
VODMFBSPGUIFDPOOFDUJPOTXSJUFTFWFSBMTJNQMF
NVMUJQMJDBUJPOFYQSFTTJPOTXJUIUIFTBNFCBTF
BOEJOUFHFSFYQPOFOUT5IFOSFXSJUFUIFTBNF
NVMUJQMJDBUJPOFYQSFTTJPOTXJUISBUJPOBMFYQPOFOUT
BOEDPNQBSFUIFTJNQMJGJDBUJPOQSPDFTTFT
1
6
exponents if the expression involves
multiplying powers with the same
base. Multiply rational exponents if the
expression involves raising a power to a
power.]
1
25. 3x 3
27. 4x 3
CC-8
3
B
4 Practice
Apply
Simplify each expression.
7
1
16
6 12
6 12
28. ; < ; <
7
7
ASSIGNMENT GUIDE
29. ;
1 2
1
30. 93 3 99
5 10
1 4
1
1 3
1 4
32. u 20 5 u6
31. m 7 m 4 7
#BTJDoPEEoo
"WFSBHFoPEEo
"EWBODFEoFWFOo
4UBOEBSEJ[FE5FTU1SFQo
.JYFE3FWJFXo
9
h 25
h 25
<
; <
13
13
33. s2 5 6s 3 5
34. Think About a Plan How would you use the Commutative Property of
Multiplication to simplify the expression in the box to the right?
+ ,
35. Reasoning Explain how the properties of integer exponents and
rational exponents are similar when raising a power to a power and
when raising a product to a power.
Mathematical Practices BSFTVQQPSUFECZ
FYFSDJTFTXJUISFEIFBEJOHT)FSFBSFUIF1SBDUJDFT
TVQQPSUFEJOUIJTMFTTPO
1
1
1 1
1
36. Error Analysis Your friend says x4 x3 x4 3 x12 . Explain the error and find the
correct answer.
C
.1.BLF4FOTFPG1SPCMFNT&Y
.13FBTPO"CTUSBDUMZ&Y
.1$SJUJRVFUIF3FBTPOJOHPG0UIFST&Y
.1"UUFOEUP1SFDJTJPO&Y
.1"UUFOEUP1SFDJTJPO&Y
Challenge
37. Reasoning What is the relationship between y and z such that xy xz 1 for all
nonzero values of x?
Standardized Test Prep
1
SAT/ACT
1
38. Simplify (6)3 (6)3 .
HOMEWORK QUICK CHECK
0
5PDIFDLTUVEFOUTVOEFSTUBOEJOHPGLFZTLJMMTBOE
DPODFQUTHPPWFS&YFSDJTFTBOE
1
=6
6
1
39. Which expression is equivalent to (6y 3)6 ?
1
y2
Short
Response
1 19
1 1
66 y2
19
6 6y 6
y6
1
1
40. Explain why the expression (a 2)2 a shows that a 2 is the same as a.
Mixed Review
Simplify each expression.
4 2
41. ; <
9
1 3
42. ; <
5
43. 2(5)2
44. (3)2 (3)2
46. k 6 k 4
47. ( f 8g 9) ( f 4g)
48.(3uv 2)(6v4)(2u2v)
50. (2z)8z5
51. 52(3a)6
52. (x9)2 x5
Simplify each expression.
45. 35 310
Simplify each expression.
49. (n3p)7
4
Common Core
Answers
Practice and Problem-Solving
Exercises (continued)
28. + 67 , 3
1
30. 93
11
7
h
29. + 13
, 5
31.m 7
3
1
3
5
32. u 33. 6 5s 5 PS 6
s
34. 5PSBJTFBSBUJPOBMQPXFSUPBSBUJPOBM
QPXFSNVMUJQMZUIFFYQPOFOUT4P*
3 917
OFFEUPNVMUJQMZ 888
917 4 *DPVME
VTFUIF$PNNVUBUJWF1SPQFSUZPG
.VMUJQMJDBUJPOUPSFPSEFSUIFGBDUPST
BOETJNQMJGZUIFGSBDUJPOTCFGPSF
NVMUJQMZJOH
35. "OTXFSTNBZWBSZ4BNQMF
BOTXFS*OUFHFSTBSFBTVCTFUPGUIF
TFUPGSBUJPOBMOVNCFSTXIJDIJTB
TVCTFUPGUIFTFUPGSFBMOVNCFST5IF
1SPQFSUJFTPG*OUFHFS&YQPOFOUTBSF
EFSJWFEGSPNUIF1SPQFSUJFTPG3FBM
/VNCFST4JODFUIFTFUPGJOUFHFSTJTB
TVCTFUPGUIFTFUPGSBUJPOBMOVNCFST
UIFTBNFFYQPOFOUQSPQFSUJFTBQQMZ
1
5
4
CC-8
36. 5PNVMUJQMZQPXFSTXJUIUIFTBNF
CBTFZPVNVTUBEEUIFFYQPOFOUT5P
BEEUIFTFFYQPOFOUTZPVNVTUGJSTU
GJOEBDPNNPOEFOPNJOBUPSOPUBEE
7
UIFEFOPNJOBUPST5IFBOTXFSJTx 1 37. 'PSUIFQSPEVDUUPFRVBMUIFTVNPG
UIFJOUFHFSTNVTUCF[FSP5IJTPDDVST
XIFO y z
Standardized Test Prep
38.#
39.(
40. + a
Z
1
,
a
1
a a
1
a a
Mixed Review
1
41. 16
42. 15
81
43.
44.
45.15PS
46.k 10
47. f 4g 10
48. 36u 3v 7
49. n 1p 7
50. 56z 13
51. 185a 6
52.x