Review: Laws of Exponentsnmnm aaa

nmn

m

aa

a

nmnm aa

n

nn

b

a

b

a

nnn baba )(

nn

aa

1

QuestionsQ: 40 =?A: 1

Q: 41 =?A: 4

Q: 41/2=? A: Let’s square the number (41/2)2 =?

(41/2)2 = 41 = 4

Recall: b is called a square root of the number a if b2 = a.

41/2 is a square root of the number 4

Or 4421

Definition of Rational Exponent

Q: 81/3=? A: Let’s cube the number (81/3)3 =?

(81/3)3 = 81 = 8So, we have

In general, for any positive integer n,

aa 2

1

33

1

88

nn aa 1

ExamplesEvaluate the following expressions

3

1

27

2

1

100

4

1

16

4

1

4

1

)16(

3

1

)8(

ExamplesSimplify the following expressions

4

4/1

2/1

n

m

44/1

42/1

)(

)(

n

m

)4()4/1(

)4()2/1(

n

m

nm2

1

2

n

m

33/123/1 )2( xyx 323/13/1 )2( yx

320 )2( yx 323 )(2 y 68y

n

m1

2

Question

4/316Which way do you prefer?

?8 3/2 23 8

3/123/2

23/13/2

)8(8

)8(8

or

22 4

3 64 43 28

4 316 ＝ 34 16

Definition of Rational Exponent

The numerator m will the exponentThe denominator n will be the index of the radical

253/2 =

n mmnn

m

aaa

2532 33

)5(25 125

Examples

1007/2 =

(-27)2/3 =

3/2

n mmnn

m

aaa

10072 7100 710 000,000,10

2723 23 9

36

1

36

1 32

3

6

1

216

1

ExamplesSimplify the following expressions

5/31032a 5/3105/3 )()32( a 63

5 32 a

68a63)2( a

2/34 )4( c 2/342/3 )(4 c

)()4( 2

343

c 632 c 68c

ExamplesSimplify the following expressions

3/29368 cba 3/293/233/263/2 )()()()8( cba

62423 8 cba 6244 cba62422 cba

3/23)27( a 3/233/2

3/2

)()27(

1

a3/2

327

1

a

223 27

1

a

22)3(

1

a 29

1

a