5.2 Properties of 5.2 Properties of Rational ExponentsRational Exponents

(p. 175)(p. 175)

Learning ObjectivesLearning Objectives

I will be able to…I will be able to…►Write Expressions in Radical FormWrite Expressions in Radical Form►Write Expressions in Rational Write Expressions in Rational Exponential FormExponential Form►Simplify and Solve Rational Exponent Simplify and Solve Rational Exponent ExpressionsExpressions

Review of Properties of Review of Properties of Exponents from section 6.1Exponents from section 6.1

►aamm ·· aann = a = am+nm+n

►(a(amm))nn = a = amnmn

►(ab)(ab)mm = a = ammbbmm

►aa00 =1 =1►aa-m-m = =► = a= am-nm-n

► ==

n

m

a

ama

1

m

b

a

m

m

b

a

These all work These all work for fraction for fraction

exponents as exponents as well as integer well as integer

exponents.exponents.

Ex: Simplify. (no decimal answers)Ex: Simplify. (no decimal answers)

a.a. 661/21/2 · 6 · 61/31/3

= 6= 61/2 + 1/31/2 + 1/3

= 6= 63/6 + 2/63/6 + 2/6

= 6= 65/65/6

b. (27b. (271/31/3 · 6 · 61/41/4))22

= (27= (271/31/3))22 · (6 · (61/41/4))22

= (3)= (3)22 · 6 · 62/42/4

= 9 · 6= 9 · 61/21/2

c.c. (4(433 · 2 · 233))-1/3-1/3

= (4= (433))-1/3-1/3 · (2 · (233))-1/3-1/3

= 4= 4-1-1 · 2 · 2-1-1

= = ¼¼ ·· ½½

= = 11//88

d.d.

= = == = =

3

4

1

4

1

9

18

3

4

1

9

18

3

4

1

2

4

3

2

** All of these examples were in rational exponent form to begin with, so the answers should be in the same form!

Laws of Rational Laws of Rational ExponentsExponents

RULESNo Negative ExponentsNo Rational Exponents

Ex: Simplify.Ex: Simplify.

a.a. ==

= = 5= = 5

b.b. = =

= = 2= = 2

Ex: Write the expression Ex: Write the expression in simplest form.in simplest form.

a.a. = == =

==

b.b. = =

= = = =

==

33 525 3 5253 125

3

3

4

323

4

32

3 8

4 64 4 416 44 416

4 42

4

8

74

4

8

7 Can’t have a tent in the basement!

4

4

4

4

2

2

8

7

4

4

16

14

2

144

** If the problem is ** If the problem is in radical form to in radical form to begin with, the begin with, the answer should be in answer should be in radical form as well.radical form as well.

Ex: Perform the indicated operationEx: Perform the indicated operation

a.a. 5(45(43/43/4) – 3(4) – 3(43/43/4))

= 2(4= 2(43/43/4))

b. b.

==

==

= =

c. c.

==

==

= = 33 381 33 3327

33 333 3 32

33 5625 33 55125

33 555 3 5 6

If the original problem is in radical form,

the answer should be in radical form as well.

If the problem is in rational exponent form, the answer should be in rational exponent form.

More ExamplesMore Examples

a.a.

b.b.

c.c.

d. d.

2x x

6 6x x

11 11y y

4 8r 4 44 rr 4 44 4 rr

rr 2r

Ex: Simplify the Expression. Ex: Simplify the Expression. Assume all variables are positive.Assume all variables are positive.

a.a.

b. (16gb. (16g44hh22))1/2 1/2

= 16= 161/21/2gg4/24/2hh2/22/2

= 4g= 4g22hh

c. c.

3 927z 3 93 27 z 33z

510

5

y

x5 10

5 5

y

x

2y

x

d.34

1

3

2

6

18

tr

rs33

2

4

11

3 tsr

33

2

4

3

3 tsr

Ex: Write the expression in Ex: Write the expression in simplest form. Assume all variables simplest form. Assume all variables

are positive.are positive.a. a.

b.

No tents in the basement!

c.

** Remember, solutions must be in the same form as the ** Remember, solutions must be in the same form as the original problem (radical form or rational exponent form)!!original problem (radical form or rational exponent form)!!

d. d. 4

6

118

z

yx4

26

2118

zz

zyx

Can’t have a tent in the basement!!

48

2118

z

zyx

2

4 2322

z

zyyx

Ex: Perform the indicated operation. Ex: Perform the indicated operation. Assume all variables are positive.Assume all variables are positive.

a.a. xx 38 x5

b. 4

1

4

1

63 ghgh 4

1

3gh

c. 44 5 662 xxx 44 662 xxxx 4 63 xx

d. sss 26 s126 s5

e. 323 7 6263 yyy

3232 6263 yyyy

32 65 yy

SolveSolve

HomeworkHomeworkTextbook pg. 178 #7-14, 29-34