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Integrated Math 2 Sections 2-7 and 2-8

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The document discusses exponents and scientific notation. It defines exponential form as representing multiplying a number by itself numerous times, with the base as the number being multiplied and the exponent as the number of times. It provides examples of evaluating expressions using exponent rules like the power, product, and quotient rules. It also demonstrates writing numbers in scientific notation and evaluating expressions for given variable values.

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SECTIONS 2-7 AND 2-8 Properties of Exponents and Zero and Negative Exponents
ESSENTIAL QUESTIONS How do you choose appropriate units of measure? How do you evaluate variable expressions? How do you write numbers using zero and negative integers as exponents? How do you write numbers in scientific notation? Where you’ll see this: Biology, finance, computers, population, physics, astronomy
VOCABULARY 1. Exponential Form: 2. Base: 3. Exponent: 4. Standard Form: 5. Scientific Notation:
VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: 3. Exponent: 4. Standard Form: 5. Scientific Notation:
VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: The number that is being multiplied over and over 3. Exponent: 4. Standard Form: 5. Scientific Notation:
VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: The number that is being multiplied over and over 3. Exponent: The number of times we multiply the base 4. Standard Form: 5. Scientific Notation:
VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: The number that is being multiplied over and over 3. Exponent: The number of times we multiply the base 4. Standard Form: Any number in decimal form 5. Scientific Notation:
VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: The number that is being multiplied over and over 3. Exponent: The number of times we multiply the base 4. Standard Form: Any number in decimal form 5. Scientific Notation: A number with two factors, where the first factor is a number ≥ 1 and < 10, and the second is a power of 10.
ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: 64 Seventh hour: Eighth hour: How long until there are 2000?
ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: 64 Seventh hour: 128 Eighth hour: How long until there are 2000?
ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: 64 Seventh hour: 128 Eighth hour: 256 How long until there are 2000?
ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: 64 Seventh hour: 128 Eighth hour: 256 How long until there are 2000? 11th hour
EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y
EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2
EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = 3(.64)(1.2)
EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = 3(.64)(1.2) = 2.304
EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = [4(.8)] (1.2) 3 2 = 3(.64)(1.2) = 2.304
EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = [4(.8)] (1.2) 3 2 = 3(.64)(1.2) = (3.2) (1.2) 3 2 = 2.304
EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = [4(.8)] (1.2) 3 2 = 3(.64)(1.2) = (3.2) (1.2) 3 2 = 2.304 = (32.768)(1.44)
EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = [4(.8)] (1.2) 3 2 = 3(.64)(1.2) = (3.2) (1.2) 3 2 = 2.304 = (32.768)(1.44) = 47.18592
PRODUCT RULE b ib = b m n m+n
PRODUCT RULE b ib = b m n m+n x ix = 3 5
PRODUCT RULE b ib = b m n m+n x ix = xixix 3 5
PRODUCT RULE b ib = b m n m+n x ix = xixix ixixixixix 3 5
PRODUCT RULE b ib = b m n m+n x ix = xixix ixixixixix = x 3 5 8
POWER RULE (b ) = b m n mn
POWER RULE (b ) = b m n mn 2 3 (y )
POWER RULE (b ) = b m n mn (y ) = y iy iy 2 3 2 2 2
POWER RULE (b ) = b m n mn (y ) = y iy iy = y 2 3 2 2 2 6
POWER OF A PRODUCT RULE (ab) = a b m m m
POWER OF A PRODUCT RULE (ab) = a b m m m 4 (gf )
POWER OF A PRODUCT RULE (ab) = a b m m m (gf ) = gf igf igf igf 4
POWER OF A PRODUCT RULE (ab) = a b m m m (gf ) = gf igf igf igf = g f 4 4 4
QUOTIENT RULE m b b ÷b = n =b m n m−n b
QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c 3 c
QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = c cicic
QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = c cicic
QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = c cicic
QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = c cicic
QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = =c 2 c cicic
POWER OF A QUOTIENT RULE m ⎛ a⎞ m a ⎜ b ⎟ = bm ⎝ ⎠
POWER OF A QUOTIENT RULE m ⎛ a⎞ m a ⎜ b ⎟ = bm ⎝ ⎠ 3 ⎛ d⎞ ⎜ w⎟ ⎝ ⎠
POWER OF A QUOTIENT RULE m ⎛ a⎞ m a ⎜ b ⎟ = bm ⎝ ⎠ 3 ⎛ d⎞ d d d ⎜ w⎟ = wiwiw ⎝ ⎠
POWER OF A QUOTIENT RULE m ⎛ a⎞ a m ⎜ b ⎟ = bm ⎝ ⎠ 3 ⎛ d⎞ d d d d 3 ⎜ w ⎟ = w i w i w = w3 ⎝ ⎠
EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m )
EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5
EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5 =3 7
EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5 =3 7 = 2187
EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5 =6 m 2 4( 2) =3 7 = 2187
EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5 =6 m 2 4( 2) =3 7 = 36m 8 = 2187
EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y
EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( 2)() 1 2 2 3
EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() 1 2 2 2 3 = ( )( ) 1 4 2 3
EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() 1 2 2 2 3 = ( )( ) 1 4 2 3 = 2 12
EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() 1 2 2 2 3 = ( )( ) 1 4 2 3 = 2 12 = 1 6
EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 2 12 = 1 6
EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 3 ( )( ) 1 8 4 9 = 2 12 = 1 6
EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 3 ( )( ) 1 8 4 9 = 2 12 = 3( 4 72 ) = 1 6
EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 3 ( )( )1 8 4 9 = 2 12 = 3( 4 72 ) = 1 6 = 12 72
EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 3 ( )( )1 8 4 9 = 2 12 = 3( 4 72 ) = 1 6 = 12 72 = 1 6
ZERO PROPERTY OF EXPONENTS b =1 0
ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4 x
ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4− 4 4 = x x
ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4− 4 4 = x =x 0 x
ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4− 4 4 = x =x 0 x 4 x 4 x
ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4− 4 4 = x =x 0 x 4 x 4 =1 x
PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b
PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h 7 h
PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h 7 = h 3−7 h
PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h −4 7 = h =h 3−7 h
PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h −4 7 = h =h 3−7 h 3 h 7 h
PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h −4 7 = h =h 3−7 h 3 h hihih 7 = h hihihihihihih
PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h −4 7 = h =h 3−7 h 3 h hihih 1 7 = = 4 h hihihihihihih h
EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y
EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y =x 2− 8
EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y =x 2− 8 −6 =x
EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y =x 2− 8 −6 =x 1 = 6 x
EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 y =x 2− 8 = 4 y −6 =x 1 = 6 x
EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 y =x 2− 8 = 4 y −6 =x =y −1 1 = 6 x
EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 y =x 2− 8 = 4 y −6 =x =y −1 1 1 = 6 = x y
EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 −6 =x 2− 8 = 4 y =z y −6 =x =y −1 1 1 = 6 = x y
EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 −6 =x 2− 8 = 4 y =z y =x −6 1 =y −1 = 6 1 1 z = 6 = x y
EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n
EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n = 6(−2) 4
EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n = 6(−2) 4 = 6(16)
EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n = 6(−2) 4 = 6(16) = 96
EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 = 6(−2) 4 =n = 6(16) = 96
EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 = 6(−2) 4 =n = 6(16) 1 = 6 n = 96
EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 = 6(−2) 4 =n = 6(16) 1 = 6 n = 96 1 = 6 4
EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 = 6(−2) 4 =n = 6(16) 1 = 6 n = 96 1 1 = 6 = 4 4096
EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 −3 = 6(−2) 4 =n = (−2) (4) 5 = 6(16) 1 = 6 n = 96 1 1 = 6 = 4 4096
EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 −3 = 6(−2) 4 =n = (−2) (4) 5 = 6(16) 1 (−2) 5 = 6 = 3 n (4) = 96 1 1 = 6 = 4 4096
EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 −3 = 6(−2) 4 =n = (−2) (4) 5 = 6(16) 1 (−2) 5 = 6 = 3 n (4) = 96 1 1 −32 = 6 = = 4 4096 64
EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 −3 = 6(−2) 4 =n = (−2) (4) 5 = 6(16) 1 (−2) 5 = 6 = 3 n (4) = 96 1 1 −32 −1 = 6 = = = 4 4096 64 2
EXAMPLE 6 Write in scientific notation. a. .0000013 b. 230,000,000,000
EXAMPLE 6 Write in scientific notation. a. .0000013 b. 230,000,000,000 -6 =1.3i10
EXAMPLE 6 Write in scientific notation. a. .0000013 b. 230,000,000,000 -6 11 =1.3i10 =2.3i10
EXAMPLE 7 Write in standard form. 6 −9 a. 7.2i10 b. 3.5i10
EXAMPLE 7 Write in standard form. 6 −9 a. 7.2i10 b. 3.5i10 = 7, 200,000
EXAMPLE 7 Write in standard form. 6 −9 a. 7.2i10 b. 3.5i10 = 7, 200,000 = .0000000035
EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms.
EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms. −24 15 (1.67i10 )(2.7i10 )
EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms. −24 15 (1.67i10 )(2.7i10 ) −24 = (1.67i2.7)(10 i10 15 )
EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms. −24 15 (1.67i10 )(2.7i10 ) −24 = (1.67i2.7)(10 i10 15 ) −9 = 4.509i10
EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms. −24 15 (1.67i10 )(2.7i10 ) −24 = (1.67i2.7)(10 i10 15 ) −9 = 4.509i10 = .000000004509 gram
HOMEWORK
HOMEWORK p. 84 #1-48 multiples of 3; p. 88 #1-48 multiples of 3 “When you’re screwing up and nobody’s saying anything to you anymore, that means they gave up [on you]...When you see yourself doing something badly and nobody’s bothering to tell you anymore, that’s a very bad place to be. Your critics are the ones who are telling you they still love and care.” - Randy Pausch

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Integrated Math 2 Sections 2-7 and 2-8

  • 1. SECTIONS 2-7 AND 2-8 Properties of Exponents and Zero and Negative Exponents
  • 2. ESSENTIAL QUESTIONS How do you choose appropriate units of measure? How do you evaluate variable expressions? How do you write numbers using zero and negative integers as exponents? How do you write numbers in scientific notation? Where you’ll see this: Biology, finance, computers, population, physics, astronomy
  • 3. VOCABULARY 1. Exponential Form: 2. Base: 3. Exponent: 4. Standard Form: 5. Scientific Notation:
  • 4. VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: 3. Exponent: 4. Standard Form: 5. Scientific Notation:
  • 5. VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: The number that is being multiplied over and over 3. Exponent: 4. Standard Form: 5. Scientific Notation:
  • 6. VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: The number that is being multiplied over and over 3. Exponent: The number of times we multiply the base 4. Standard Form: 5. Scientific Notation:
  • 7. VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: The number that is being multiplied over and over 3. Exponent: The number of times we multiply the base 4. Standard Form: Any number in decimal form 5. Scientific Notation:
  • 8. VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: The number that is being multiplied over and over 3. Exponent: The number of times we multiply the base 4. Standard Form: Any number in decimal form 5. Scientific Notation: A number with two factors, where the first factor is a number ≥ 1 and < 10, and the second is a power of 10.
  • 9. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
  • 10. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
  • 11. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
  • 12. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
  • 13. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
  • 14. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
  • 15. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
  • 16. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: 64 Seventh hour: Eighth hour: How long until there are 2000?
  • 17. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: 64 Seventh hour: 128 Eighth hour: How long until there are 2000?
  • 18. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: 64 Seventh hour: 128 Eighth hour: 256 How long until there are 2000?
  • 19. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: 64 Seventh hour: 128 Eighth hour: 256 How long until there are 2000? 11th hour
  • 20. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y
  • 21. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2
  • 22. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = 3(.64)(1.2)
  • 23. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = 3(.64)(1.2) = 2.304
  • 24. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = [4(.8)] (1.2) 3 2 = 3(.64)(1.2) = 2.304
  • 25. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = [4(.8)] (1.2) 3 2 = 3(.64)(1.2) = (3.2) (1.2) 3 2 = 2.304
  • 26. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = [4(.8)] (1.2) 3 2 = 3(.64)(1.2) = (3.2) (1.2) 3 2 = 2.304 = (32.768)(1.44)
  • 27. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = [4(.8)] (1.2) 3 2 = 3(.64)(1.2) = (3.2) (1.2) 3 2 = 2.304 = (32.768)(1.44) = 47.18592
  • 28. PRODUCT RULE b ib = b m n m+n
  • 29. PRODUCT RULE b ib = b m n m+n x ix = 3 5
  • 30. PRODUCT RULE b ib = b m n m+n x ix = xixix 3 5
  • 31. PRODUCT RULE b ib = b m n m+n x ix = xixix ixixixixix 3 5
  • 32. PRODUCT RULE b ib = b m n m+n x ix = xixix ixixixixix = x 3 5 8
  • 33. POWER RULE (b ) = b m n mn
  • 34. POWER RULE (b ) = b m n mn 2 3 (y )
  • 35. POWER RULE (b ) = b m n mn (y ) = y iy iy 2 3 2 2 2
  • 36. POWER RULE (b ) = b m n mn (y ) = y iy iy = y 2 3 2 2 2 6
  • 37. POWER OF A PRODUCT RULE (ab) = a b m m m
  • 38. POWER OF A PRODUCT RULE (ab) = a b m m m 4 (gf )
  • 39. POWER OF A PRODUCT RULE (ab) = a b m m m (gf ) = gf igf igf igf 4
  • 40. POWER OF A PRODUCT RULE (ab) = a b m m m (gf ) = gf igf igf igf = g f 4 4 4
  • 41. QUOTIENT RULE m b b ÷b = n =b m n m−n b
  • 42. QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c 3 c
  • 43. QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = c cicic
  • 44. QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = c cicic
  • 45. QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = c cicic
  • 46. QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = c cicic
  • 47. QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = =c 2 c cicic
  • 48. POWER OF A QUOTIENT RULE m ⎛ a⎞ m a ⎜ b ⎟ = bm ⎝ ⎠
  • 49. POWER OF A QUOTIENT RULE m ⎛ a⎞ m a ⎜ b ⎟ = bm ⎝ ⎠ 3 ⎛ d⎞ ⎜ w⎟ ⎝ ⎠
  • 50. POWER OF A QUOTIENT RULE m ⎛ a⎞ m a ⎜ b ⎟ = bm ⎝ ⎠ 3 ⎛ d⎞ d d d ⎜ w⎟ = wiwiw ⎝ ⎠
  • 51. POWER OF A QUOTIENT RULE m ⎛ a⎞ a m ⎜ b ⎟ = bm ⎝ ⎠ 3 ⎛ d⎞ d d d d 3 ⎜ w ⎟ = w i w i w = w3 ⎝ ⎠
  • 52. EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m )
  • 53. EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5
  • 54. EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5 =3 7
  • 55. EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5 =3 7 = 2187
  • 56. EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5 =6 m 2 4( 2) =3 7 = 2187
  • 57. EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5 =6 m 2 4( 2) =3 7 = 36m 8 = 2187
  • 58. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y
  • 59. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( 2)() 1 2 2 3
  • 60. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() 1 2 2 2 3 = ( )( ) 1 4 2 3
  • 61. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() 1 2 2 2 3 = ( )( ) 1 4 2 3 = 2 12
  • 62. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() 1 2 2 2 3 = ( )( ) 1 4 2 3 = 2 12 = 1 6
  • 63. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 2 12 = 1 6
  • 64. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 3 ( )( ) 1 8 4 9 = 2 12 = 1 6
  • 65. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 3 ( )( ) 1 8 4 9 = 2 12 = 3( 4 72 ) = 1 6
  • 66. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 3 ( )( )1 8 4 9 = 2 12 = 3( 4 72 ) = 1 6 = 12 72
  • 67. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 3 ( )( )1 8 4 9 = 2 12 = 3( 4 72 ) = 1 6 = 12 72 = 1 6
  • 68. ZERO PROPERTY OF EXPONENTS b =1 0
  • 69. ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4 x
  • 70. ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4− 4 4 = x x
  • 71. ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4− 4 4 = x =x 0 x
  • 72. ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4− 4 4 = x =x 0 x 4 x 4 x
  • 73. ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4− 4 4 = x =x 0 x 4 x 4 =1 x
  • 74. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b
  • 75. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h 7 h
  • 76. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h 7 = h 3−7 h
  • 77. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h −4 7 = h =h 3−7 h
  • 78. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h −4 7 = h =h 3−7 h 3 h 7 h
  • 79. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h −4 7 = h =h 3−7 h 3 h hihih 7 = h hihihihihihih
  • 80. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h −4 7 = h =h 3−7 h 3 h hihih 1 7 = = 4 h hihihihihihih h
  • 81. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y
  • 82. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y =x 2− 8
  • 83. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y =x 2− 8 −6 =x
  • 84. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y =x 2− 8 −6 =x 1 = 6 x
  • 85. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 y =x 2− 8 = 4 y −6 =x 1 = 6 x
  • 86. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 y =x 2− 8 = 4 y −6 =x =y −1 1 = 6 x
  • 87. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 y =x 2− 8 = 4 y −6 =x =y −1 1 1 = 6 = x y
  • 88. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 −6 =x 2− 8 = 4 y =z y −6 =x =y −1 1 1 = 6 = x y
  • 89. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 −6 =x 2− 8 = 4 y =z y =x −6 1 =y −1 = 6 1 1 z = 6 = x y
  • 90. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n
  • 91. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n = 6(−2) 4
  • 92. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n = 6(−2) 4 = 6(16)
  • 93. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n = 6(−2) 4 = 6(16) = 96
  • 94. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 = 6(−2) 4 =n = 6(16) = 96
  • 95. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 = 6(−2) 4 =n = 6(16) 1 = 6 n = 96
  • 96. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 = 6(−2) 4 =n = 6(16) 1 = 6 n = 96 1 = 6 4
  • 97. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 = 6(−2) 4 =n = 6(16) 1 = 6 n = 96 1 1 = 6 = 4 4096
  • 98. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 −3 = 6(−2) 4 =n = (−2) (4) 5 = 6(16) 1 = 6 n = 96 1 1 = 6 = 4 4096
  • 99. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 −3 = 6(−2) 4 =n = (−2) (4) 5 = 6(16) 1 (−2) 5 = 6 = 3 n (4) = 96 1 1 = 6 = 4 4096
  • 100. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 −3 = 6(−2) 4 =n = (−2) (4) 5 = 6(16) 1 (−2) 5 = 6 = 3 n (4) = 96 1 1 −32 = 6 = = 4 4096 64
  • 101. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 −3 = 6(−2) 4 =n = (−2) (4) 5 = 6(16) 1 (−2) 5 = 6 = 3 n (4) = 96 1 1 −32 −1 = 6 = = = 4 4096 64 2
  • 102. EXAMPLE 6 Write in scientific notation. a. .0000013 b. 230,000,000,000
  • 103. EXAMPLE 6 Write in scientific notation. a. .0000013 b. 230,000,000,000 -6 =1.3i10
  • 104. EXAMPLE 6 Write in scientific notation. a. .0000013 b. 230,000,000,000 -6 11 =1.3i10 =2.3i10
  • 105. EXAMPLE 7 Write in standard form. 6 −9 a. 7.2i10 b. 3.5i10
  • 106. EXAMPLE 7 Write in standard form. 6 −9 a. 7.2i10 b. 3.5i10 = 7, 200,000
  • 107. EXAMPLE 7 Write in standard form. 6 −9 a. 7.2i10 b. 3.5i10 = 7, 200,000 = .0000000035
  • 108. EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms.
  • 109. EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms. −24 15 (1.67i10 )(2.7i10 )
  • 110. EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms. −24 15 (1.67i10 )(2.7i10 ) −24 = (1.67i2.7)(10 i10 15 )
  • 111. EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms. −24 15 (1.67i10 )(2.7i10 ) −24 = (1.67i2.7)(10 i10 15 ) −9 = 4.509i10
  • 112. EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms. −24 15 (1.67i10 )(2.7i10 ) −24 = (1.67i2.7)(10 i10 15 ) −9 = 4.509i10 = .000000004509 gram
  • 114. HOMEWORK p. 84 #1-48 multiples of 3; p. 88 #1-48 multiples of 3 “When you’re screwing up and nobody’s saying anything to you anymore, that means they gave up [on you]...When you see yourself doing something badly and nobody’s bothering to tell you anymore, that’s a very bad place to be. Your critics are the ones who are telling you they still love and care.” - Randy Pausch