How do we handle fractional exponents?

Do Now:28 =

2? =26 =2? =2? =

22 =21 =

2? =

2? =

256

128643225 =16

4

24 =

27 =

2

1/4

1/8

2-2 =

2-3 =

Fill in the appropriateinformation

23 = 8

2-1 = 1/2

How do we handle fractional exponents?

Do Now:Simplify/Rationalize:

5 2 1

2 2 1

Properties of Exponents

Zero Power Property a0 = 1

Product of Powers Property am • an = am+n

Power of Power Property (am)n = am•n

Negative Power Property a-n = 1/an, a 0

Power of Product Property (ab)m = ambm

Quotients of Powers Property

am

an am n , a 0

Power of Quotient Property

(a

b)m

am

bm , b 0

Indices, Exponents, and New Power Rules

Product of Powers Property am • an = am+n

Power of Product Property (ab)m = am • bm

Power of Quotient Property

a

b

m

am

bm

example: 82 • 83 = 82 + 3 = 85

example: (2 • 8)2 = 22 • 82

example: x3 • x6 = x3 + 6 = x9

example: (xy)5 = x5 • y5

example:

2

3

2

22

32

x

y

2

x2

y2

Types of exponents

Positive Integer Exponent an = a • a • a • • • • an factors

Rational Exponent 1/n

a1 n an

Rational Expo. m/n

am n (a1 n)m ( an )m

3 2

0 ( )

. ( 2)

m n mnIf a and n is even a a

results in a nonreal number

ex imaginary number

beware! -23/2 is not the same as (-2)3/2

Negative Exponent -n

a n 1

an

Zero Exponent a0 = 10

- m/n

a ( m n ) 1

am nNegativeRational Exponent

Roots, Radicals & Rational Exponents

Square Root

anradical sign

radicand

index

of a number is one of the twoequal factors whose product is that number

Every positive real number has two square roots

The principal square root of a positive number k is itspositive square root, .

k

has an index of 2

k2 kcan be written exponentially as

k2 k k1 2

81 9

81 9

0 0

kIf k < 0, is an imaginarynumber

811

2 9

Cube Root

Roots, Radicals & Rational Exponents

anradical sign

radicand

index

of a number is one of the threeequal factors whose product is that number

( k3 )( k3 )( k3 ) k

k3has an index of 3

273 3 273 3

principal cube roots

can be written exponentially ask1/3

k3 k1 3

271

3 3

Roots, Radicals & Rational Exponents

anradical sign

radicand

index

nth Rootof a number is one of nequal factors whose product is that number

k5 k1 5can be written exponentially ask1/n

325 2 325 2

principal odd roots

646 2 646 not real

principal even roots

has an index where n is any counting number

kn

641

6 2

square root -

nth root -

kn = k1/n

cube root -

( k3 )( k3 )( k3 ) k

k3 k1 3

Indices and Rational Exponents

k2 k

k2 k k1 2

k1/2 • k1/2 = k 1/2 + 1/2 = k1 = k

k1/3 • k1/3 • k1/3 = k 1/3 + 1/3 + 1/3 = k1 = k

k1/n • k1/n • k1/n . . . = k 1/n + 1/n + 1/n. . . = k1 = k

n times n times

21/2 • 21/2 = 2 1/2 + 1/2 = 21 = 2

21/3 • 21/3 • 21/3 = 2 1/3 + 1/3 + 1/3 = 21 = 2

81/3 • 81/3 • 81/3 = 8 1/3 + 1/3 + 1/3 = 81 = 8

Fractional ExponentsRadicals

Fractional Exponents

23 2

baba (ab)m = am • bm

3 2 32 4

rational exponentpositive integer exponent

multiplication law

Simplifying

x9 2x1 2

(x1 3)6

x8x2

(x2)2

x (82)

(x2)2

= x6

x (9 21 2)

(x1 3 )6

= x3

division law

simplify &power law

x10 2

x2 x5

x2

= x5 - 2

x10

x4

= x10 - 4

Simplifying – Fractional Exponents

A rational expression that contains a fractional exponent in the denominator must also be rationalized. When you simplify an expression, be sure your answer meets all of the given conditions.

Conditions for a Simplified Expression1. It has no negative exponents.2. It has no fractional exponents in the

denominator.3. It is not a complex fraction.4. The index of any remaining radical is

as small as possible.

Model Problems

1 2 13 3 21) 2) 3) (3 )a y a

Rewrite using radicals:

334) 3 5) 2 5 6) 2a

Rewrite using rational exponents:

1 1 5

3 2 29

7) ( 125) 8) ( ) 9) 416

Evaluate:

Evaluating

Evaluate a0 + a1/3 + a -2 when a = 8

80 + 81/3 + 8-2 replace a with 8

1 + 81/3 + 8-2 x0 = 1

x1/3 =

x3

83 21 + 2 + 8-2

x–n = 1/xn 8–2 = 1/82 = 1/641 + 2 + 1/64

3 1/64 combine like terms

If m = 8, find the value of (8m0)2/3

(8 • 80)2/3 replace m with 8

(8)2/3

(8 • 1)2/3 x0 = 1

= 4

Simplifying – Fractional Exponents

3 5

4 4m n

1 1 5

3 6 12a b c

3 42 2