a1/n
If is a real number, then 1 / .n na an a
Slide 10.2- 3
Use exponential notation for nth roots.
Notice that the denominator of the rational exponent is the index of the radical.
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 10.2- 4
CLASSROOM EXAMPLE 1
Evaluating Exponentials of the Form a1/n
Solution:
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 10.2- 6
Define and use expressions of the form am/n.
Evaluate each exponential.
3/ 225 31/ 225
2/327
325
23 27 9
35 125
21/327 23
Slide 10.2- 7
CLASSROOM EXAMPLE 2
Evaluating Exponentials of the Form am/n
Solution:
3/ 216 31/ 216 316 34 64
2/364 23 64 16 21/3
64 24
3/ 236 is not a real number, since or is not a real number.
1/ 236 , 36,
a–m/n
If am/n is a real number, then 1
0 //
( ).m nm n
a aa
Slide 10.2- 8
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.
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
31
36
3
1
6
1
216
3/ 225
64
3
25
64
35
8
125
512
Slide 10.2- 9
CLASSROOM EXAMPLE 3
Evaluating Exponentials with Negative Rational Exponents
Solution:
am/n
If all indicated roots are real numbers, then
11 // / .
m nm n n ma a a
Slide 10.2- 10
Define and use expressions of the form am/n.
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 10.2- 11
Define and use expressions of the form am/n.
Write each exponential as a radical. Assume that all variables, represent positive real numbers.
1/ 219 12 119 9 3/ 411 34 112/314x 2314 x
3/53/55 2x x 3 35 55 2x x
Slide 10.2- 13
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
371/ 237
84 9 4 28/ 89 9 1
8 8z = |z|
Write each radical as an exponential.
Slide 10.2- 14
CLASSROOM EXAMPLE 4
Converting between Rational Exponents and Radicals (cont’d)
Solution:
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
a
r r
r
a b
b a
r r
r
a a
b b
1
r
raa
Slide 10.2- 16
Use the rules for exponents with rational exponents.
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 10.2- 17
CLASSROOM EXAMPLE 5
Applying Rules for Rational Exponents
Solution:
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 10.2- 18
CLASSROOM EXAMPLE 5
Applying Rules for Rational Exponents (cont’d)
Write with only positive exponents. Assume that all variables represent positive real numbers.
Solution:
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 10.2- 19
CLASSROOM EXAMPLE 6
Applying Rules for Rational Exponents
Solution: