int math 2 section 2-7/2-8 1011
DESCRIPTION
Properties of Exponents and Zero and Negative ExponentsTRANSCRIPT
SECTIONS 2-7 AND 2-8Properties 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 base4. 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 base4. Standard Form: Any number in decimal form5. 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 base4. Standard Form: Any number in decimal form5. 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.
ORGANISM ACTIVITY
Hours 1 2 3 4 5
# of organisms
2 4 8 16 32
Pattern:
Sixth hour: Seventh hour: Tenth hour:
How long until there are 2000?
An organism divides into two new organisms each hour. Fill in the table for the number of organisms after each hour.
ORGANISM ACTIVITY
Hours 1 2 3 4 5
# of organisms
2 4 8 16 32
Pattern:
Sixth hour: Seventh hour: Tenth hour:
How long until there are 2000?
An organism divides into two new organisms each hour. Fill in the table for the number of organisms after each hour.
ORGANISM ACTIVITY
Hours 1 2 3 4 5
# of organisms
2 4 8 16 32
Pattern:
Sixth hour: Seventh hour: Tenth hour:
How long until there are 2000?
An organism divides into two new organisms each hour. Fill in the table for the number of organisms after each hour.
ORGANISM ACTIVITY
Hours 1 2 3 4 5
# of organisms
2 4 8 16 32
Pattern:
Sixth hour: Seventh hour: Tenth hour:
How long until there are 2000?
An organism divides into two new organisms each hour. Fill in the table for the number of organisms after each hour.
ORGANISM ACTIVITY
Hours 1 2 3 4 5
# of organisms
2 4 8 16 32
Pattern:
Sixth hour: Seventh hour: Tenth hour:
How long until there are 2000?
An organism divides into two new organisms each hour. Fill in the table for the number of organisms after each hour.
ORGANISM ACTIVITY
Hours 1 2 3 4 5
# of organisms
2 4 8 16 32
Pattern:
Sixth hour: Seventh hour: Tenth hour:
How long until there are 2000?
An organism divides into two new organisms each hour. Fill in the table for the number of organisms after each hour.
ORGANISM ACTIVITY
Hours 1 2 3 4 5
# of organisms
2 4 8 16 32
Pattern: The number of organisms doubles
Sixth hour: Seventh hour: Tenth hour:
How long until there are 2000?
An organism divides into two new organisms each hour. Fill in the table for the number of organisms after each hour.
ORGANISM ACTIVITY
Hours 1 2 3 4 5
# of organisms
2 4 8 16 32
Pattern: The number of organisms doubles
Sixth hour: 64 Seventh hour: Tenth hour:
How long until there are 2000?
An organism divides into two new organisms each hour. Fill in the table for the number of organisms after each hour.
ORGANISM ACTIVITY
Hours 1 2 3 4 5
# of organisms
2 4 8 16 32
Pattern: The number of organisms doubles
Sixth hour: 64 Seventh hour: 128 Tenth hour:
How long until there are 2000?
An organism divides into two new organisms each hour. Fill in the table for the number of organisms after each hour.
ORGANISM ACTIVITY
Hours 1 2 3 4 5
# of organisms
2 4 8 16 32
Pattern: The number of organisms doubles
Sixth hour: 64 Seventh hour: 128 Tenth hour: 1024
How long until there are 2000?
An organism divides into two new organisms each hour. Fill in the table for the number of organisms after each hour.
ORGANISM ACTIVITY
Hours 1 2 3 4 5
# of organisms
2 4 8 16 32
Pattern: The number of organisms doubles
Sixth hour: 64 Seventh hour: 128 Tenth hour: 1024
How long until there are 2000? 11th hour
An organism divides into two new organisms each hour. Fill in the table for the number of organisms after each hour.
EXAMPLE 1
Evaluate for x = .8 and y = 1.2.
a. 3x2 y b. (4x)3 y2
EXAMPLE 1
Evaluate for x = .8 and y = 1.2.
a. 3x2 y b. (4x)3 y2
3(.8)2(1.2)
EXAMPLE 1
Evaluate for x = .8 and y = 1.2.
a. 3x2 y b. (4x)3 y2
3(.8)2(1.2)
3(.64)(1.2)
EXAMPLE 1
Evaluate for x = .8 and y = 1.2.
a. 3x2 y b. (4x)3 y2
3(.8)2(1.2)
3(.64)(1.2)
2.304
EXAMPLE 1
Evaluate for x = .8 and y = 1.2.
a. 3x2 y b. (4x)3 y2
3(.8)2(1.2)
3(.64)(1.2)
2.304
[4(.8)]3(1.2)2
EXAMPLE 1
Evaluate for x = .8 and y = 1.2.
a. 3x2 y b. (4x)3 y2
3(.8)2(1.2)
3(.64)(1.2)
2.304
[4(.8)]3(1.2)2
(3.2)3(1.2)2
EXAMPLE 1
Evaluate for x = .8 and y = 1.2.
a. 3x2 y b. (4x)3 y2
3(.8)2(1.2)
3(.64)(1.2)
2.304
[4(.8)]3(1.2)2
(3.2)3(1.2)2
(32.768)(1.44)
EXAMPLE 1
Evaluate for x = .8 and y = 1.2.
a. 3x2 y b. (4x)3 y2
3(.8)2(1.2)
3(.64)(1.2)
2.304
[4(.8)]3(1.2)2
(3.2)3(1.2)2
(32.768)(1.44)
47.18592
PRODUCT RULE
bmibn = bm+n
PRODUCT RULE
bmibn = bm+n
x3 ix5 =
PRODUCT RULE
bmibn = bm+n
x3 ix5 = xixix
PRODUCT RULE
bmibn = bm+n
x3 ix5 = xixix ixixixixix
PRODUCT RULE
bmibn = bm+n
x3 ix5 = xixix ixixixixix = x8
POWER RULE
(bm )n = bmn
POWER RULE
(bm )n = bmn
(y2 )3
POWER RULE
(bm )n = bmn
(y2 )3 = y2 iy2 iy2
POWER RULE
(bm )n = bmn
(y2 )3 = y2 iy2 iy2
= y6
POWER OF A PRODUCT RULE
(ab)m = ambm
POWER OF A PRODUCT RULE
(ab)m = ambm
(gf )4
POWER OF A PRODUCT RULE
(ab)m = ambm
(gf )4 = gf igf igf igf
POWER OF A PRODUCT RULE
(ab)m = ambm
(gf )4 = gf igf igf igf = g4 f 4
QUOTIENT RULE
bm ÷ bn =
bm
bn = bm−n
QUOTIENT RULE
bm ÷ bn =
bm
bn = bm−n
c5
c3
QUOTIENT RULE
bm ÷ bn =
bm
bn = bm−n
c5
c3 =
ciciciciccicic
QUOTIENT RULE
bm ÷ bn =
bm
bn = bm−n
c5
c3 =
ciciciciccicic
QUOTIENT RULE
bm ÷ bn =
bm
bn = bm−n
c5
c3 =
ciciciciccicic
QUOTIENT RULE
bm ÷ bn =
bm
bn = bm−n
c5
c3 =
ciciciciccicic
QUOTIENT RULE
bm ÷ bn =
bm
bn = bm−n
c5
c3 =
ciciciciccicic = c2
POWER OF A QUOTIENT RULE
ab
⎛⎝⎜
⎞⎠⎟
m
=am
bm
POWER OF A QUOTIENT RULE
ab
⎛⎝⎜
⎞⎠⎟
m
=am
bm
dw
⎛⎝⎜
⎞⎠⎟
3
POWER OF A QUOTIENT RULE
ab
⎛⎝⎜
⎞⎠⎟
m
=am
bm
dw
⎛⎝⎜
⎞⎠⎟
3
=
dw
idw
idw
POWER OF A QUOTIENT RULE
ab
⎛⎝⎜
⎞⎠⎟
m
=am
bm
dw
⎛⎝⎜
⎞⎠⎟
3
=
dw
idw
idw
=d3
w3
EXAMPLE 2
Simplify.
a. 32 i35 b. (6m4 )2
EXAMPLE 2
Simplify.
a. 32 i35 b. (6m4 )2
32+5
EXAMPLE 2
Simplify.
a. 32 i35 b. (6m4 )2
32+5
37
EXAMPLE 2
Simplify.
a. 32 i35 b. (6m4 )2
32+5
37
2187
EXAMPLE 2
Simplify.
a. 32 i35 b. (6m4 )2
32+5
37
2187
62 m4(2)
EXAMPLE 2
Simplify.
a. 32 i35 b. (6m4 )2
32+5
37
2187
62 m4(2)
36m8
EXAMPLE 3
Evaluate for x = 1/2 and y = 2/3.
a. x2 y b. 3x3 y2
EXAMPLE 3
Evaluate for x = 1/2 and y = 2/3.
a. x2 y b. 3x3 y2
12( )2 2
3( )
EXAMPLE 3
Evaluate for x = 1/2 and y = 2/3.
a. x2 y b. 3x3 y2
12( )2 2
3( )
14( ) 2
3( )
EXAMPLE 3
Evaluate for x = 1/2 and y = 2/3.
a. x2 y b. 3x3 y2
12( )2 2
3( )
14( ) 2
3( )
212
EXAMPLE 3
Evaluate for x = 1/2 and y = 2/3.
a. x2 y b. 3x3 y2
12( )2 2
3( )
14( ) 2
3( )
212
16
EXAMPLE 3
Evaluate for x = 1/2 and y = 2/3.
a. x2 y b. 3x3 y2
12( )2 2
3( )
14( ) 2
3( )
212
16
312( )3 2
3( )2
EXAMPLE 3
Evaluate for x = 1/2 and y = 2/3.
a. x2 y b. 3x3 y2
12( )2 2
3( )
14( ) 2
3( )
212
16
312( )3 2
3( )2
318( ) 4
9( )
EXAMPLE 3
Evaluate for x = 1/2 and y = 2/3.
a. x2 y b. 3x3 y2
12( )2 2
3( )
14( ) 2
3( )
212
16
312( )3 2
3( )2
318( ) 4
9( ) 3
472( )
EXAMPLE 3
Evaluate for x = 1/2 and y = 2/3.
a. x2 y b. 3x3 y2
12( )2 2
3( )
14( ) 2
3( )
212
16
312( )3 2
3( )2
318( ) 4
9( ) 3
472( )
1272
EXAMPLE 3
Evaluate for x = 1/2 and y = 2/3.
a. x2 y b. 3x3 y2
12( )2 2
3( )
14( ) 2
3( )
212
16
312( )3 2
3( )2
318( ) 4
9( ) 3
472( )
1272
16
ZERO PROPERTY OF EXPONENTS
b0 = 1
ZERO PROPERTY OF EXPONENTS
b0 = 1
x4
x4
ZERO PROPERTY OF EXPONENTS
b0 = 1
x4
x4 = x4−4
ZERO PROPERTY OF EXPONENTS
b0 = 1
x4
x4 = x4−4 = x0
ZERO PROPERTY OF EXPONENTS
b0 = 1
x4
x4 = x4−4 = x0
x4
x4
ZERO PROPERTY OF EXPONENTS
b0 = 1
x4
x4 = x4−4 = x0
x4
x4 = 1
PROPERTY OF NEGATIVE EXPONENTS
b−m
=
1bm
PROPERTY OF NEGATIVE EXPONENTS
b−m
=
1bm
h3
h7
PROPERTY OF NEGATIVE EXPONENTS
b−m
=
1bm
h3
h7 = h3−7
PROPERTY OF NEGATIVE EXPONENTS
b−m
=
1bm
h3
h7 = h3−7 = h−4
PROPERTY OF NEGATIVE EXPONENTS
b−m
=
1bm
h3
h7 = h3−7 = h−4
h3
h7
PROPERTY OF NEGATIVE EXPONENTS
b−m
=
1bm
h3
h7 = h3−7 = h−4
h3
h7 =
hihihhihihihihihih
PROPERTY OF NEGATIVE EXPONENTS
b−m
=
1bm
h3
h7 = h3−7 = h−4
h3
h7 =
hihihhihihihihihih
=1h4
EXAMPLE 4
Simplify each expression. Write the answer with a positive exponent.
a. x2
x8
b. y3 i
1y4 c. (z−3 )2
EXAMPLE 4
Simplify each expression. Write the answer with a positive exponent.
a. x2
x8
b. y3 i
1y4 c. (z−3 )2
x2−8
EXAMPLE 4
Simplify each expression. Write the answer with a positive exponent.
a. x2
x8
b. y3 i
1y4 c. (z−3 )2
x2−8
x−6
EXAMPLE 4
Simplify each expression. Write the answer with a positive exponent.
a. x2
x8
b. y3 i
1y4 c. (z−3 )2
x2−8
x−6
1x6
EXAMPLE 4
Simplify each expression. Write the answer with a positive exponent.
a. x2
x8
b. y3 i
1y4 c. (z−3 )2
x2−8
x−6
1x6
y3
y4
EXAMPLE 4
Simplify each expression. Write the answer with a positive exponent.
a. x2
x8
b. y3 i
1y4 c. (z−3 )2
x2−8
x−6
1x6
y3
y4
y−1
EXAMPLE 4
Simplify each expression. Write the answer with a positive exponent.
a. x2
x8
b. y3 i
1y4 c. (z−3 )2
x2−8
x−6
1x6
y3
y4
y−1
1y
EXAMPLE 4
Simplify each expression. Write the answer with a positive exponent.
a. x2
x8
b. y3 i
1y4 c. (z−3 )2
x2−8
x−6
1x6
y3
y4
y−1
1y
z−6
EXAMPLE 4
Simplify each expression. Write the answer with a positive exponent.
a. x2
x8
b. y3 i
1y4 c. (z−3 )2
x2−8
x−6
1x6
y3
y4
y−1
1y
z−6
1z6
EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
a. 6m4 b. (n3 )−2 c. m5n−3
EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
a. 6m4 b. (n3 )−2 c. m5n−3
6(−2)4
EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
a. 6m4 b. (n3 )−2 c. m5n−3
6(−2)4
6(16)
EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
a. 6m4 b. (n3 )−2 c. m5n−3
6(−2)4
6(16)
96
EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
a. 6m4 b. (n3 )−2 c. m5n−3
6(−2)4
6(16)
96
n−6
EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
a. 6m4 b. (n3 )−2 c. m5n−3
6(−2)4
6(16)
96
n−6
1n6
EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
a. 6m4 b. (n3 )−2 c. m5n−3
6(−2)4
6(16)
96
n−6
1n6
146
EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
a. 6m4 b. (n3 )−2 c. m5n−3
6(−2)4
6(16)
96
n−6
1n6
146
1
4096
EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
a. 6m4 b. (n3 )−2 c. m5n−3
6(−2)4
6(16)
96
n−6
1n6
146
1
4096
(−2)5(4)−3
EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
a. 6m4 b. (n3 )−2 c. m5n−3
6(−2)4
6(16)
96
n−6
1n6
146
1
4096
(−2)5(4)−3
(−2)5
(4)3
EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
a. 6m4 b. (n3 )−2 c. m5n−3
6(−2)4
6(16)
96
n−6
1n6
146
1
4096
(−2)5(4)−3
(−2)5
(4)3
−3264
EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
a. 6m4 b. (n3 )−2 c. m5n−3
6(−2)4
6(16)
96
n−6
1n6
146
1
4096
(−2)5(4)−3
(−2)5
(4)3
−3264
−12
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
1.3i10-6
EXAMPLE 6
Write in scientific notation.
a. .0000013 b. 230,000,000,000
1.3i10-6 2.3i1011
EXAMPLE 7
Write in standard form.
a. 7.2i106 b. 3.5i10−9
EXAMPLE 7
Write in standard form.
a. 7.2i106 b. 3.5i10−9
7, 200,000
EXAMPLE 7
Write in standard form.
a. 7.2i106 b. 3.5i10−9
7, 200,000 .0000000035
EXAMPLE 8
The mass of one hydrogen atom is 1.67i10−24
grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms.
EXAMPLE 8
The mass of one hydrogen atom is 1.67i10−24
grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms.
(1.67i10−24 )(2.7i1015 )
EXAMPLE 8
The mass of one hydrogen atom is 1.67i10−24
grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms.
(1.67i10−24 )(2.7i1015 )
(1.67i2.7)(1015 i10−24 )
EXAMPLE 8
The mass of one hydrogen atom is 1.67i10−24
grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms.
(1.67i10−24 )(2.7i1015 )
(1.67i2.7)(1015 i10−24 )
4.509i10−9
EXAMPLE 8
The mass of one hydrogen atom is 1.67i10−24
grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms.
(1.67i10−24 )(2.7i1015 )
(1.67i2.7)(1015 i10−24 )
4.509i10−9
.000000004509 gram
PROBLEM SET
PROBLEM SET
p. 84 #1-48 multiples of 3; p. 88 #1-48 multiples of 3
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