RATIONAL EXPONENTS

Assignments

Assignments

Basic Terminology

.

,

.

,

TIONMULTIPLICA

repeatedundergoesthatVariableornumberthenotationlExponentiaInBase

factoraasusedis

basethetimesofnumberthenotationlExponentiaInExponent

42BASE

EXPONENTmeans 162222

Important Examples

IMPORTANT EXAMPLES81)3333(34 means

81)3()3()3()3()3( 4 means

27)333(33 means

27)3()3()3()3( 3 means

Variable Examples

Variable Expressions

))()((

))()()()(()(

3

5

yyymeansy

xxxxxtionmultiplicaforsparentheseusemeansx

Substitution and EvaluatingSTEPS

1.Write out the original problem.2.Show the substitution with parentheses.3.Work out the problem.

3;4: xxifSolveExample 3;4 xxifSolve

3)4( = 64

More Examples

Evaluate the variable expression when x = 1, y = 2, and w = -3

22 )()( yx

22 )()( yx

22 )2()1(

541

Step 1

Step 2

Step 3

2)( yx

2)( yx

2)2()1(

9)3( 2

Step 1

Step 2

Step 3

ywx

ywx

2)1)(3(

Step 1

Step 2

Step 3

3)1)(3(

MULTIPLICATION PROPERTIESPRODUCT OF POWERS

This property is used to combine 2 or more exponential expressions with the SAME base.

53 22 )222( )22222( 82 256

))(( 43 xx ))()(( xxx ))()()(( xxxx 7x

MULTIPLICATION PROPERTIESPOWER TO A POWER

This property is used to write and exponential expression as a single power of the base.

32 )5( )5)(5)(5( 222 65

42 )(x ))()()(( 2222 xxxx 8x

MULTIPLICATION PROPERTIESPOWER OF PRODUCT

This property combines the first 2 multiplication properties to simplify exponential expressions.

2)56( )5()6( 22 9002536

3)5( xy ))()(5( 333 yx33125 yx

532 )4( xx 5323 ))(4( xx 5222 ))()(()64( xxxx

56 ))(64( xx 1164x

MULTIPLICATION PROPERTIESSUMMARY

PRODUCT OF POWERSbaba xxx

POWER TO A POWER baba xx

POWER OF PRODUCTaaa yxxy )(

ADD THE EXPONENTS

MULTIPLY THE EXPONENTS

ZERO AND NEGATIVE EXPONENTSANYTHING TO THE ZERO POWER IS 1.

27

1

3

13

9

1

3

13

3

1

3

13

13

33

93

273

33

22

11

0

1

2

3

22222

222

4

1

2

1

)2(

1)2(

2122

xxxx

xxx

8131

31

3

11

311

3

1 44

4

4

4

DIVISION PROPERTIES QUOTIENT OF POWERS

This property is used when dividing two or more exponential expressions with the same base.

))()((

))()()()((3

5

xxx

xxxxx

x

x 2

1

))((x

xx

7434

34

3

4

3 11111

xxxx

xxx

x

x

DIVISION PROPERTIESPOWER OF A QUOTIENT

12

8

43

424

3

2

)(

)(

y

x

y

x

y

x

Hard Example

3

43

2

3

2

yx

xy

343

32

)3(

)2(

yx

xy

1293

633

3

2

yx

yx

69

123

27

8

yx

yx

69

123

27

8

yx

yx6

6

27

8

x

ySummary of Zero, Negative, and Division Properties

ZERO, NEGATIVE, AND DIVISION PROPERTIES

Zero power 1)( 0 x

Negative Exponents

aa

aa

xx

andx

x

1

1

Quotient of powers

bab

a

xx

x

Power of a quotient

a

aa

y

x

y

x