6.1 Properties of Exponents

2/4/2013

Power, Base and Exponent:

73

Exponent: is the number that tells you how many times the base is multiplied to itself.

In this example 73 means 7•7•7

Product of Powers:

Ex. 32 • 35

= 3•3•3•3•3•3•3 = 37

= 32+5

In general: am•an = am+n

Power of a Power:

Ex. (23 )2

= (23 )• (23 )=(2•2•2)•(2•2•2)= 26 = 23•2

In general: (am)n = am•n

Power of a Product:

Ex. (4•3 )3

= (4•3 )• (4•3 ) • (4•3 ) =(4•4•4)•(3•3•3) = 4 3 •3 3

In general: (a •b)m =a

m•b m

Zero Exponent

In general: a0 = 1

44 164 644 123

55 255 1255 123

33 93 273 123

÷5 ÷5 ÷5

÷4 ÷4 ÷4

÷3÷3 ÷3

Any base raised to a 0 power equals 1.

50 = 1

40 = 1

30 = 1

3

Exponential Form Fraction Form

33 27

32 9

31 3

30 1

3-1

3-2 =

3-3 =

Negative Exponent

Ex. 5-2

Ex.

In general:

25

1

mm

aa

1

Negative exponent MOVES power. If the power with a negative exponent is in the numerator, the power moves to the denominator and exponent becomes positive. If the power with a negative exponent is in the denominator, the power moves to the numerator and exponent becomes positive.

33

22

xx

Quotient of Powers

Ex.

In general:

33

33333

3

32

5

nmn

m

aa

a

253 33

Power of a Quotient

Ex.

In general:

4

44

5

3

5

3

5

3

5

3

5

3

5

3

m

mm

b

a

b

a

Example 1 Evaluate Expressions with Negative Exponents

( ) 82 – – ( )42– Product of powers property= ( ) 8 42 – – +

= ( ) 42 – – Simplify exponent.

1

( )42–= Negative exponent property

=161

Evaluate power.

Example 2 Evaluate Quotients with Exponents

33

35 2

Evaluate .

Quotient of powers property 33

35 2

= ( )232

= 34 Power of a power property

= 81 Evaluate power.

Checkpoint Evaluate Numerical Expressions

( )3221.

Evaluate the expression.

ANSWER 64

( )3502. ANSWER 1

3. ( ) 53 – –( )23– ANSWER271–

ANSWER278

4.3

2 3

Example 3 Simplify Algebraic Expressions

a.y

x– 3

2

Power of a quotient propertyx 2

( )2y – 3 =

=x 2

y – 3 2 •Power of a power property

=x 2

y – 6 Simplify exponent.

= x 2y 6 Negative exponent property

Example 3 Simplify Algebraic Expressions

= 25y y 5y – 6

Simplify exponent.

= 25y – 6 5 1 + + Product of powers property

= 25y 0 Simplify exponent.

= 25 Zero exponent property

( )25y – 3 y 5y b. = ( )2y – 3 y 5y 52 Power of a product property

= 25y y 5y – 3 2 • Power of a power property

Example 3 Simplify Algebraic Expressions

c.x 5y 2–

x 3y 6

= x – 3 5 y – 6 ( 2) – Quotient of powers property

= x 2y 8– Simplify exponent.

=y 8

x 2Negative exponent property

Checkpoint Simplify Algebraic Expressions

( )32p p 45.8p 7

6.xy 4–

x 5y 3x 4y 7

Simplify the expression.

7. ( )33b – 2 b 8 27b 2

–

s

r 2

– 4

3

8.1

r 6s 12

ANSWER

Homework:

6.1 p.299 #8-34 evenDisregard direction that

says. “Tell which properties you used”