section 2chapter 8. 1 copyright © 2012, 2008, 2004 pearson education, inc. objectives 2 3 4...
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Section 2Chapter 8
1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objectives
2
3
4
Rational Exponents
Use exponential notation for nth roots.
Define and use expressions of the form am/n.
Convert between radicals and rational exponents.
Use the rules for exponents with rational exponents.
8.2
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use exponential notation for nth roots.
Objective 1
Slide 8.2- 3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
a1/n
If is a real number, then 1 / .n na an a
Slide 8.2- 4
Use exponential notation for nth roots.
Notice that the denominator of the rational exponent is the index of the radical.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Evaluate each exponential.
1/532 5 32 2
1/ 264 2 64 64 8
1/ 481 4 81 3
1/ 4( 81) 4 81 Is not a real number because the radicand, –81, is negative and the index, 4, is even.
1/3( 64) 3 64 4
1/31
27
31
27
1
3
Slide 8.2- 5
CLASSROOM EXAMPLE 1
Evaluating Exponentials of the Form a1/n
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Define and use expressions of the form am/n.
Objective 2
Slide 8.2- 6
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
am/n
If m and n are positive integers with m/n in lowest terms, then
provided that a1/n is a real number. If a1/n is not a real number, then am/n is not a real number.
1/ / ,mm n na a
Slide 8.2- 7
Define and use expressions of the form am/n.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Evaluate each exponential.
3/ 225 31/ 225
2/327
3
25
23 27 9
35 125
21/327 23
Slide 8.2- 8
CLASSROOM EXAMPLE 2
Evaluating Exponentials of the Form am/n
Solution:
3/ 216 31/ 216 3
16 34 64
2/364 2
3 64 16 21/3
64 24
3/ 236 is not a real number, since or
is not a real number. 1/ 2
36 , 36,
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a–m/n
If am/n is a real number, then 1
0 //
( ).m nm n
a aa
Slide 8.2- 9
Define and use expressions of the form am/n.
A negative exponent does not necessarily lead to a negative result. Negative exponents lead to reciprocals, which may be positive.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Evaluate each exponential.
3/ 4813/ 4
1
81
31/ 4
1
81
3/ 264
25
34
1
81
3
1
3 1
27
3/ 2363/ 2
1
36
31/ 2
1
36
3
1
36
3
1
6
1
216
3/ 225
64
3
25
64
35
8
125
512
Slide 8.2- 10
CLASSROOM EXAMPLE 3
Evaluating Exponentials with Negative Rational Exponents
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
am/n
If all indicated roots are real numbers, then
11 // / .
m nm n n ma a a
Slide 8.2- 11
Define and use expressions of the form am/n.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Radical Form of am/n
If all indicated roots are real numbers, then
That is, raise a to the mth power and then take the nth root, or take the nth root of a and then raise to the mth power.
/ .m
nm n m na a a
Slide 8.2- 12
Define and use expressions of the form am/n.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Convert between radicals and rational exponents.
Objective 3
Slide 8.2- 13
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Write each exponential as a radical. Assume that all variables, represent positive real numbers.
1/ 219 12 119 9
3/ 411 34 11
2/314x 2314 x
3/53/55 2x x 3 35 55 2x x
Slide 8.2- 14
CLASSROOM EXAMPLE 4
Converting between Rational Exponents and Radicals
Solution:
5/ 7x
5/ 7 57
11
x x
1/32 2x y 2 23 x y
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371/ 237
84 9 4 28/ 89 9 1
8 8z = z, since z is assumed to be positive.
Write each radical as an exponential.
Slide 8.2- 15
CLASSROOM EXAMPLE 4
Converting between Rational Exponents and Radicals (cont’d)
Solution:
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Use the rules for exponents with rational exponents.
Objective 4
Slide 8.2- 16
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Rules for Rational Exponents
Let r and s be rational numbers. For all real numbers a and b for which the indicated expressions exist,
r s r sa a a
sr rsa a
1 rr
aa
r r rab a b
r
r ss
aa
b
r r
r
a b
b a
r r
r
a a
b b
1
r
raa
Slide 8.2- 17
Use the rules for exponents with rational exponents.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Write with only positive exponents. Assume that all variables represent positive real numbers.
1/ 2 1/33 3 1/ 2 1/33 2/3
4/3
7
7
3/ 6 2/ 63 5/ 63
2/3 4/3 /3
22/
37 71
7
61/3 2/3a b
b
61/3 2/3 1a b 61/3 1/3a b 6 61/3 1/3a b
1/3 1/36 6a b 6/3 6/3a b 2 2a b2
2
a
b
Slide 8.2- 18
CLASSROOM EXAMPLE 5
Applying Rules for Rational Exponents
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
1/ 23 4
2 1/5
a b
a b
1/ 23 ( 2) 4 1/5a b
2/5 3/5 8/5r r r
1/ 25 21/5a b
1/ 2 1/ 25 21/5a b 5/ 2 21/10a b
21/10
5/ 2
b
a
2/5 3/5 2 /5 8/5r r r r
2/5 3/5 2 /5 8/5r r 5/5 10/5r r 2r r Slide 8.2- 19
CLASSROOM EXAMPLE 5
Applying Rules for Rational Exponents (cont’d)
Write with only positive exponents. Assume that all variables represent positive real numbers.
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Write all radicals as exponentials, and then apply the rules for rational exponents. Leave answers in exponential form. Assume that all variables represent positive real numbers.
34 5x x 3/ 4 1/5x x 3/ 4 1/5x 15/ 20 4/ 20x 19/ 20x
5
3
x
x
5/ 25/ 2 1/3 15/ 6 2/ 6
1/13/ 6
3
xx x
xx
3 6 x 1/3 1/11/ 6 1/33 1/ 86 1/ 6x x x x
Slide 8.2- 20
CLASSROOM EXAMPLE 6
Applying Rules for Rational Exponents
Solution: