Properties of Exponents

p. 323

Properties of Exponentsa&b are real numbers, m&n are integers

• Product Property:• Quotient of Powers:

• Power of a Power Property: • Power of a Product Property:

• Negative Exponent Property:• Zero Exponent Property: • Power of Quotient:

Example – Product Property

• (-5) 3 (-5) 2 = • (-5)(-5)(-5)(-5)(-5)=• (-5) 5

• (-5)3+2 =• (-5) 5

Example – Product Property

• x5 • x2 = x•x•x•x•x•x•x

• x5+2 =

• x7

Product Property a&b are real numbers, m&n are integers

• Product Property: (am )(an)=am+n

• a3• a5 • a4 =

• a3• a5 • a4 = a3+5+4

• a3• a5 • a4 = a12

Product Property

• (a3 b2) (a4 b6) = • (a3 a4) (b2 b6) = a3+4 b2+6

• a3+4 b2+6 = a7 b8

• (x5 y2) (x4 y7) = • (x5 x4) (y2 y7) = x5+4 y2+7

• x5+2 y2+7 = x9 y9

You try

• (3x6 y4) (4xy7) =

• (3x6 y4) (4xy7) = (3•4)x6+1 • y4+7

• (3•4)x6+1 • y4+7 = 12x7y11

• (2x12 y5) (6x3 y9) =

• (2• 6)x12+3 y5+9 =12x15y14

Do now

• (2x4 y4) (5xy7) =

• (2x4 y4) (5xy7) = (2•5)x4+1 • y4+7

• (2•5)x4+1 • y4+7 = 10x5y11

• (3x14 y5) (9x3 y) =

• (3• 9)x14+3 y5+1 =27x17y6

Dividing Powers with Like bases

• -5 3 = -5• -5• -5-5 2 -5• -5

• -5• -5• -5 = -5

-5• -5

Power of a Quotient with like bases

• x 4 = x• x • x• x

X2 x• x

• X2

Quotient of Powers

3

5

x

x 35x 2x

Quotient of Powers

• Quotient of Powers:

• am = am-n; a≠0

an

You try

• 45x4y7 = 43x2y6

• 45x4y7 = 45-3 x4-2y7-6

43x2y6

• 45-3 x4-2y7-6 = 42x2y = 16x2y

You try

• 37x9y12 = 34x5y6

• 37x9y12 = 37-4 x9-5 y12-6

34x5y6

• 37-4 x9-5 y12-6= 33x4y8 = 27x4y8

Negative Exponents

• x 2 = x• x_____ x4 x• x • x• x• 1 = x2

• x 2 = x 2 -4 = x-2

X4

x-2 = 1

x2

Negative exponets

• x 3 = x• x_ • x___ x5 x• x • x• x • x• 1 = x3

• x 3 = x 3 -5 = x-3

x5

• x-3 = 1 x3

Example – Quotient of Powers

10

5

x

x 105x 5x 5

1

x

You try

• x-2 =• 1 x2

2x-2y = 2x-2y = 2y x2

You try

• (-5)-6(-5)4 =

• (-5)-6+4 =

• (-5)-2 =

25

1

25

1

Properties of Exponentsa&b are real numbers, m&n are integers

• Negative Exponent Property:

• a-m= ; a≠0ma

1

Zero Exponent Property

• x0

• x2 = x2-2

x2

x2-2 = x0

• x2 = 1

x2

x0= 1

You try

(x-2) (x2) =

(x-2) (x2) = x-2+2

x-2+2 = x0

x0 = 1

Properties of Exponentsa&b are real numbers, m&n are integers ets Review

• Zero Exponent Property: a0=1; a≠0

Properties of Exponentsa&b are real numbers, m&n are integers

• Product Property: am * an=am+n

• Quotient of Powers: am = am-n; a≠0 an

• Negative Exponent Property: a-m= ; a≠0

• Zero Exponent Property: a0=1; a≠0

ma

1

Journal Entry:Describe the rules for the follwoing

• Product Property: • Quotient of Powers:

• Negative Exponent Property:• Zero Exponent Property:

Example – Power of a Power

• (23)4 = (23) (23) (23) (23)4

• 23+3+3+3 =

• (23)4 = 212

Example - Power of a Power

• (34)3 = (34) (34) (34)

• (34) (34) (34) = 34+4+4

• (34)3 = 312

Raising a Power to a Power

•(X5)2 = (X5) (X5)

•(X5) (X5)= x5+5

•(X5)2 = x10

Power of a Power Property a&b are real numbers, m&n are integers

• Power of a Power Property: (am)n=amn

• (x5)3 = x5•3

• x5•3= x15

You try

• (y4)8 =

• (y4)8 = y4•8 = y24

• (s3)4 =

• (s3)4 = s3•4 = s12

Power of a Product Property

• (-2x7)2 = (-2x7) (-2x7)

• (-2x7) (-2x7) = (-2• -2) (x7 •x7) =4x14

• (-2x7)2 = (-2)2 (x7)2 = -21•2 x7•2

• = 4x14

• Power of a Product Property: (ab)m=ambm

• (a3b2)4= (a3)4 (b2)4

• (a3)4 (b2)4 = a3•4b2•4 =a12b8

You try

• (-2x4)3 = (-2)1•3 x5•3

• (-2x4)3 = (-2)1•3 x4•3

• (-2)1•3 x4•3 = (-2)3 x12 = -16x12

• (4x4y5)2

• (4x4y5)2 = 41•2x4 •2y5•2

• (4x4y5)2 = 16x8y10

You try

• (-3x5y3)4 =

• (-3)1•4 x5•4 y4•4= (-3)4 x20 y16

• (7x3y-5)2

• 71•2x3 •2y-5•2

• 16x8y-10 = 16x8

• y10

Example – Power of Quotient

2

5s

r

25

2

s

r 10

2

s

r 102sr

Properties of Exponentsa&b are real numbers, m&n are integers

• Power of Quotient: • b≠0 m

mm

b

a

b

a

Properties of Exponentsa&b are real numbers, m&n are integers

• Product Property: am * an=am+n

• Quotient of Powers: am = am-n; a≠0 an

Power of a Power Property: (am)n=amn

• Power of a Product Property: (ab)m=ambm

• Negative Exponent Property: a-m= ; a≠0

• Zero Exponent Property: a0=1; a≠0• Power of Quotient: b≠0

m

mm

b

a

b

a

ma

1

Multiplying and Dividing Monomials• Monomial – an expression that is either

a numeral, a variable or a product of numerals and variables with whole number exponents.

• Constant – Monomial that is a numeral. Example - 2

Journal Entry:Describe the rules for the follwoing

Power of a Power Property:

• Power of a Product Property:

• Power of Quotient:

Multiplying Monomials

• (-2x4y2) (-3xy2z3) =

• (-2)(-3)(x4x )(y2y2 ) z3

• 6x5y4 z3

• (-2x3y4) 2 (-3xy2)

• (-2 ) 2 x3∙2y4∙2 ) (-3xy2)

• (4)(-3)(x6x) (y8 y2 )

• -12x7y10

Scientific Notation

• A number is expressed in scientific notation when it is written as the product of a factor and a power of 10. The factor must be greater than or equal to 1 and less then 10

• a x 10ⁿ, where 1 ≤ a < 10

Scientific Notation

• 131,400,000,000=

1.314 x 1011

Move the decimal behind the 1st number

How many places did you have to move the decimal?

Put that number here!

Write using scientific notation

• 12,300=

• 1.23 x 104

• Write using standard notation

• 1.76 x 103

• 1,760

Example – Scientific Notation

• 131,400,000,000 =• 5,284,000

1.314 x 1011 =

5.284 x 106

61110*284.5

314.1 900,2410*249. 5

Example – Scientific Notation

• (5.2 x 109)(3.0 x 10-3 )=

• (5.2 x 3.0) (109 x 10-3 )=

• 15.6 x 106

• 1.56 x 107

• 2.45 x 10-3 =

• 0.00245

Properties of Exponentsa&b are real numbers, m&n are integers

• Product Property: am * an=am+n

• Quotient of Powers: am = am-n; a≠0 an

Power of a Power Property: (am)n=amn

• Power of a Product Property: (ab)m=ambm

• Negative Exponent Property: a-m= ; a≠0

• Zero Exponent Property: a0=1; a≠0• Power of Quotient: b≠0

m

mm

b

a

b

a

ma

1