int math 2 section 2-7/2-8 1011

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:


VOCABULARY


2. Base: The number that is being multiplied over and over


VOCABULARY



3. Exponent: The number of times we multiply the base4. Standard Form:5. Scientific Notation:

VOCABULARY



3. Exponent: The number of times we multiply the base4. Standard Form: Any number in decimal form5. Scientific Notation:

VOCABULARY



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: The number of organisms doubles

Sixth hour: Seventh hour: Tenth hour:



ORGANISM ACTIVITY

Hours 1 2 3 4 5

# of organisms

2 4 8 16 32


Sixth hour: 64 Seventh hour: Tenth hour:



ORGANISM ACTIVITY

Hours 1 2 3 4 5

# of organisms

2 4 8 16 32


Sixth hour: 64 Seventh hour: 128 Tenth hour:



ORGANISM ACTIVITY

Hours 1 2 3 4 5

# of organisms

2 4 8 16 32


Sixth hour: 64 Seventh hour: 128 Tenth hour: 1024



ORGANISM ACTIVITY

Hours 1 2 3 4 5

# of organisms

2 4 8 16 32


Sixth hour: 64 Seventh hour: 128 Tenth hour: 1024

How long until there are 2000? 11th hour


EXAMPLE 1

Evaluate for x = .8 and y = 1.2.

a. 3x2 y b. (4x)3 y2

EXAMPLE 1


a. 3x2 y b. (4x)3 y2

3(.8)2(1.2)

EXAMPLE 1


a. 3x2 y b. (4x)3 y2

3(.8)2(1.2)

3(.64)(1.2)

EXAMPLE 1


a. 3x2 y b. (4x)3 y2

3(.8)2(1.2)

3(.64)(1.2)

2.304

EXAMPLE 1


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


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


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


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


(ab)m = ambm

(gf )4


(ab)m = ambm

(gf )4 = gf igf igf igf


(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 = c2

POWER OF A QUOTIENT RULE

ab

⎛⎝⎜

⎞⎠⎟

m

=am

bm


ab

⎛⎝⎜

⎞⎠⎟

m

=am

bm

dw

⎛⎝⎜

⎞⎠⎟

3


ab

⎛⎝⎜

⎞⎠⎟

m

=am

bm

dw

⎛⎝⎜

⎞⎠⎟

3

=

dw

idw

idw


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


a. x2 y b. 3x3 y2

12( )2 2

3( )

EXAMPLE 3


a. x2 y b. 3x3 y2

12( )2 2

3( )

14( ) 2

3( )

EXAMPLE 3


a. x2 y b. 3x3 y2

12( )2 2

3( )

14( ) 2

3( )

212

EXAMPLE 3


a. x2 y b. 3x3 y2

12( )2 2

3( )

14( ) 2

3( )

212

16

EXAMPLE 3


a. x2 y b. 3x3 y2

12( )2 2

3( )

14( ) 2

3( )

212

16

312( )3 2

3( )2

EXAMPLE 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


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


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


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


b0 = 1

x4

x4


b0 = 1

x4

x4 = x4−4


b0 = 1

x4

x4 = x4−4 = x0


b0 = 1

x4

x4 = x4−4 = x0

x4

x4


b0 = 1

x4

x4 = x4−4 = x0

x4

x4 = 1

PROPERTY OF NEGATIVE EXPONENTS

b−m

=

1bm


b−m

=

1bm

h3

h7


b−m

=

1bm

h3

h7 = h3−7


b−m

=

1bm

h3

h7 = h3−7 = h−4


b−m

=

1bm

h3

h7 = h3−7 = h−4

h3

h7


b−m

=

1bm

h3

h7 = h3−7 = h−4

h3

h7 =

hihihhihihihihihih


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


a. x2

x8

b. y3 i

1y4 c. (z−3 )2

x2−8

EXAMPLE 4


a. x2

x8

b. y3 i

1y4 c. (z−3 )2

x2−8

x−6

EXAMPLE 4


a. x2

x8

b. y3 i

1y4 c. (z−3 )2

x2−8

x−6

1x6

EXAMPLE 4


a. x2

x8

b. y3 i

1y4 c. (z−3 )2

x2−8

x−6

1x6

y3

y4

EXAMPLE 4


a. x2

x8

b. y3 i

1y4 c. (z−3 )2

x2−8

x−6

1x6

y3

y4

y−1

EXAMPLE 4


a. x2

x8

b. y3 i

1y4 c. (z−3 )2

x2−8

x−6

1x6

y3

y4

y−1

1y

EXAMPLE 4


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


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


a. 6m4 b. (n3 )−2 c. m5n−3

6(−2)4

EXAMPLE 5


a. 6m4 b. (n3 )−2 c. m5n−3

6(−2)4

6(16)

EXAMPLE 5


a. 6m4 b. (n3 )−2 c. m5n−3

6(−2)4

6(16)

96

EXAMPLE 5


a. 6m4 b. (n3 )−2 c. m5n−3

6(−2)4

6(16)

96

n−6

EXAMPLE 5


a. 6m4 b. (n3 )−2 c. m5n−3

6(−2)4

6(16)

96

n−6

1n6

EXAMPLE 5


a. 6m4 b. (n3 )−2 c. m5n−3

6(−2)4

6(16)

96

n−6

1n6

146

EXAMPLE 5


a. 6m4 b. (n3 )−2 c. m5n−3

6(−2)4

6(16)

96

n−6

1n6

146

1

4096

EXAMPLE 5


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


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


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


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


a. .0000013 b. 230,000,000,000

1.3i10-6

EXAMPLE 6


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


a. 7.2i106 b. 3.5i10−9

7, 200,000

EXAMPLE 7


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



(1.67i10−24 )(2.7i1015 )

EXAMPLE 8



(1.67i10−24 )(2.7i1015 )

(1.67i2.7)(1015 i10−24 )

EXAMPLE 8



(1.67i10−24 )(2.7i1015 )

(1.67i2.7)(1015 i10−24 )

4.509i10−9

EXAMPLE 8



(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

“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

int math 2 section 2-7/2-8 1011

Education

number of times

new organisms

organism divides

organism activity

exponential form

standard form

decimal form

scientic notation