In parts (g) – (i), write each radical as an exponential. Simplify. Assume that all vari-
ables represent positive real numbers.
(g) (h)
(i) 26 z^6 =z,since zis positive. NOW TRY
210 = 10 1/2 2438 = 3 8/4 = 32 = 9
SECTION 8.2 Rational Exponents 439
NOW TRY
EXERCISE 4
Write each exponential as a
radical. Assume that all
variables represent positive
real numbers.
(a) (b)
(c)
(d) (e)
In parts (f ) – (h), write each
radical as an exponential.
Simplify. Assume that all
variables represent positive
real numbers.
(f ) (g)
(h) 24 x^4
2315 2442
w-2/5 1 a^2 - b^22 1/44 t3/5+ 14 t 2 2/321 1/2 17 5/4
NOTE InExample 4(i),it is not necessary to use absolute value bars, since the direc-
tions specifically state that the variable represents a positive real number. Because the
absolute value of the positive real number zis zitself, the answer is simply z.
OBJECTIVE 4 Use the rules for exponents with rational exponents. The
definition of rational exponents allows us to apply the rules for exponents from
Section 5.1.
Rules for Rational ExponentsLet rand sbe rational numbers. For all real numbers aand bfor which the indi-
cated expressions exist, the following are true.
ara
1
a
b
ra
a
b
b
rar
br
1 ar 2 sars 1 ab 2 rarbr
a
a
b
b
rbr
ar
ar
as
ar ars
1
ar
ar#asars
Applying Rules for Rational ExponentsWrite with only positive exponents. Assume that all variables represent positive real
numbers.
(a)
Product rule= 2 3/4 Add exponents.
= 2 1/2+1/4
2 1/2# 2 1/4
EXAMPLE 5
(c)
Power rulePower ruleQuotient rule
8
3 -^1 =8
3 -3
3 =5= x 3
(^2) y5/3
= x^2 y8/3-^1
=
x^2 y8/3
y^1
=
1 x1/2 241 y2/3 24
y
1 x1/2^ y2/3 24
y
NOW TRY ANSWERS
- (a) (b)
(c)(d) (e)(f ) 15 1/3 (g) 2 (h)x24 a^2 - b^2
1
A^25 w^ B
24 A 25 t B
3
+A 234 t B
2A^2417 B
221 5(b)
Quotient rule
Subtract exponents.