Properties Of Exponents

Properties of Exponents p. 323

Properties of Exponents a&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 x 5 • x 2 = x •x• x •x•x•x•x x 5+2 = x 7

Product Property a&b are real numbers, m&n are integers Product Property : (a m )(a n )=a m+n a 3 • a 5 • a 4 = a 3 • a 5 • a 4 = a 3+5+4 a 3 • a 5 • a 4 = a 12

Product Property (a 3 b 2 ) (a 4 b 6 ) = (a 3 a 4 ) (b 2 b 6 ) = a 3+4 b 2+6 a 3+4 b 2+6 = a 7 b 8 (x 5 y 2 ) (x 4 y 7 ) = (x 5 x 4 ) (y 2 y 7 ) = x 5+4 y 2+7 x 5+2 y 2+7 = x 9 y 9

You try (3x 6 y 4 ) (4xy 7 ) = (3x 6 y 4 ) (4xy 7 ) = (3 •4) x 6+1 • y 4+7 (3 •4) x 6+1 • y 4+7 = 12x 7 y 11 (2x 12 y 5 ) (6x 3 y 9 ) = (2 • 6) x 12+3 y 5+9 =12x 15 y 14

Do now (2x 4 y 4 ) (5xy 7 ) = (2x 4 y 4 ) (5xy 7 ) = (2 •5) x 4+1 • y 4+7 (2 •5) x 4+1 • y 4+7 = 10x 5 y 11 (3x 14 y 5 ) (9x 3 y) = (3 • 9) x 14+3 y 5+1 =27x 17 y 6

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 X 2 x • x X 2

Quotient of Powers Quotient of Powers : a m = a m-n ; a≠0 a n

You try 4 5 x 4 y 7 = 4 3 x 2 y 6 4 5 x 4 y 7 = 4 5-3 x 4-2 y 7-6 4 3 x 2 y 6 4 5-3 x 4-2 y 7-6 = 4 2 x 2 y = 16x 2 y

You try 3 7 x 9 y 12 = 3 4 x 5 y 6 3 7 x 9 y 12 = 3 7-4 x 9-5 y 12-6 3 4 x 5 y 6 3 7-4 x 9-5 y 12-6 = 3 3 x 4 y 8 = 27x 4 y 8

Negative Exponents x 2 = x • x_____ x 4 x • x • x • x 1 = x 2 x 2 = x 2 -4 = x -2 X 4 x -2 = 1 x 2

Negative exponets x 3 = x • x_ • x___ x 5 x • x • x • x • x 1 = x 3 x 3 = x 3 -5 = x -3 x 5 x -3 = 1 x 3

Example – Quotient of Powers

You try x -2 = 1 x 2 2x -2 y = 2x -2 y = 2y x 2

You try (-5) -6 (-5) 4 = (-5) -6+4 = (-5) -2 =

Properties of Exponents a&b are real numbers, m&n are integers Negative Exponent Property : a -m = ; a ≠0

Zero Exponent Property x 0 x 2 = x 2-2 x 2 x 2-2 = x 0 x 2 = 1 x 2 x 0 = 1

You try (x -2 ) (x 2 ) = (x -2 ) (x 2 ) = x -2+2 x -2+2 = x 0 x 0 = 1

Properties of Exponents a&b are real numbers, m&n are integers ets Review Zero Exponent Property : a 0 =1; a≠0

Properties of Exponents a&b are real numbers, m&n are integers Product Property : a m * a n =a m+n Quotient of Powers : a m = a m-n ; a≠0 a n Negative Exponent Property : a -m = ; a ≠0 Zero Exponent Property : a 0 =1; a≠0

Journal Entry: Describe the rules for the follwoing Product Property : Quotient of Powers : Negative Exponent Property : Zero Exponent Property :

Example – Power of a Power (2 3 ) 4 = (2 3 ) (2 3 ) (2 3 ) (2 3 ) 4 2 3+3+3+3 = (2 3 ) 4 = 2 12

Example - Power of a Power (3 4 ) 3 = (3 4 ) (3 4 ) (3 4 ) (3 4 ) (3 4 ) (3 4 ) = 3 4+4+4 (3 4 ) 3 = 3 12

Raising a Power to a Power (X 5 ) 2 = (X 5 ) (X 5 ) (X 5 ) (X 5 )= x 5+5 (X 5 ) 2 = x 10

Power of a Power Property a&b are real numbers, m&n are integers Power of a Power Property : (a m ) n =a mn (x 5 ) 3 = x 5 • 3 x 5 • 3 = x 15

You try (y 4 ) 8 = (y 4 ) 8 = y 4 •8 = y 24 (s 3 ) 4 = (s 3 ) 4 = s 3 •4 = s 12

Power of a Product Property (-2x 7 ) 2 = (-2x 7 ) (-2x 7 ) (-2x 7 ) (-2x 7 ) = (-2 • - 2) (x 7 • x 7 ) =4x 14 (-2x 7 ) 2 = (-2) 2 (x 7 ) 2 = -2 1 •2 x 7 • 2 = 4x 14

Power of a Product Property : (ab) m =a m b m (a 3 b 2 ) 4 = (a 3 ) 4 (b 2 ) 4 (a 3 ) 4 (b 2 ) 4 = a 3 •4 b 2 •4 =a 12 b 8

You try (-2x 4 ) 3 = (-2) 1 • 3 x 5 • 3 (-2x 4 ) 3 = (-2) 1 • 3 x 4 • 3 (-2) 1 • 3 x 4 • 3 = (-2) 3 x 12 = -16x 12 (4x 4 y 5 ) 2 (4x 4 y 5 ) 2 = 4 1 • 2 x 4 • 2 y 5 • 2 (4x 4 y 5 ) 2 = 16x 8 y 10

You try (-3x 5 y 3 ) 4 = (-3) 1 • 4 x 5 • 4 y 4 • 4 = (-3) 4 x 20 y 16 (7x 3 y -5 ) 2 7 1 • 2 x 3 • 2 y -5 • 2 16x 8 y -10 = 16x 8 y 10

Properties of Exponents a&b are real numbers, m&n are integers Power of Quotient: b≠0

Properties of Exponents a&b are real numbers, m&n are integers Product Property : a m * a n =a m+n Quotient of Powers : a m = a m-n ; a≠0 a n Power of a Power Property : (a m ) n =a mn Power of a Product Property : (ab) m =a m b m Negative Exponent Property : a -m = ; a ≠0 Zero Exponent Property : a 0 =1; a≠0 Power of Quotient: b≠0

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 (-2x 4 y 2 ) (-3xy 2 z 3 ) = (-2)(-3)(x 4 x )(y 2 y 2 ) z 3 6x 5 y 4 z 3 (-2x 3 y 4 ) 2 (-3xy 2 ) (-2 ) 2 x 3∙2 y 4∙2 ) (-3xy 2 ) (4)(-3)(x 6 x) (y 8 y 2 ) -12x 7 y 10

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 10 11 Move the decimal behind the 1 st number How many places did you have to move the decimal? Put that number here!

Write using scientific notation 12,300= 1.23 x 10 4 Write using standard notation 1.76 x 10 3 1,760

Example – Scientific Notation 131,400,000,000 = 5,284,000 1.314 x 10 11 = 5.284 x 10 6

Example – Scientific Notation (5.2 x 10 9 )(3.0 x 10 -3 )= (5.2 x 3.0) (10 9 x 10 -3 )= 15.6 x 10 6 1.56 x 10 7 2.45 x 10 -3 = 0.00245

Properties Of Exponents

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Properties Of Exponents