Download Ch. 7 Notes

Algebra 1 Chapter 7 Notes
Chapter 7: Exponents
Name: ___________________
Section 7.1: Negative and Zero Exponents
A. Review of Exponents
baseexponent
The _______________ represents the number of times the __________ is multiplied by itself.
The exponent applies to only ________ number or symbol unless parenthesis are used.
2x2 = 2(___)(___) = _____
versus (2x)2 = (____)(____) = _____
Evaluate:
1.) 33
2.) 25
3.) -44
5.) -24
6.) (-2)4
7.) 3x2y3 if x = -2 and y = -1
B. Simplifying Zero Exponents
Explore:
x5
8.) 5
x
9.)
4.)
x 2 y8
x 2 y8
10.)
34
34
**For every non-zero number a, a0 = _____**
Evaluate:
11.) 60
12.) -30
C. Simplifying Negative Exponents
Explore:
x3
x2 y6
15.) 5
16.) 3 8
x
x y
13.) 6xy0
14.)
17.)
2
40
33
35
1
Algebra 1 Chapter 7 Notes
**When I have a ___________ exponent, I make the exponent ___________
and move that term only to the _________________________________.**
a-n 
1
,
a __
1
 a __
-n
a
Simplify. Express using positive exponents:
18.) x-3
21.)
22
6x 4
19.) 4m-2
20.) 3-2
22.) x-3y4
23.)
3a 2b2
c0
Evaluate each expression if: x = 2 and y = -3
 Re-write each expression using positive exponents
 Plug in values of x and y. Evaluate.
24.) x2y-3
27.)
1
x y3
2
25.) 3-2x-2y2
26.) x-yyx
28.) 2x0y-2
29.) 4x-2
2
Algebra 1 Chapter 7 Notes
Name:_______________________
Notes #1: Sections 7.1 – 7.3
Section 7.1: Review
Simplify; leave all answers in positive exponents:
1.) m-3
2.) y-1
3.) 6m0
4.) 2-3
5.) 4-2
6.) -3-3
7.) (7m4)0
8.)
11.) a-1 + b-1
12.) (ab)-2
3x 0
22 y 3
Evaluate if a = -2 and b = -3:
9.) a2b-1
10.) 2ab-1
Section 7.2: Scientific Notation
A. Why Scientific Notation?
Compare:
145,000,000 vs 1.45 x 108
Why do we use scientific notation?
0.00000023
vs 2.3 x 10-7
When will it be used most often?
B. Format for Scientific Notation
A number in scientific notation is written as the product of two factors in the form
a x 10n, where n is an integer and 1  a  10
GOAL: ___ . ___ ___ ___ x 10 ___
Examples:
-2.7 x 10-3
3.081 x 1011
Is each number written in scientific notation? If no, explain.
1.) 35 x 107
2.) -3.903 x 101
3.) -12 x 100.5
3
Algebra 1 Chapter 7 Notes
C. Writing in Standard Notation
Converting from Scientific Notation to Standard Notation
•
If 10 is to a ________ power, move the decimal that many places to the ________
•
If 10 is to a ________ power, move the decimal that many places to the ________
Write in standard notation:
-4
4.) 2.35 x 103
5.) -1.4 x 10
6.) 9.876 x 10
-6
7.) -5.61 x 10
8
D. Writing in Scientific Notation
Converting from Standard Notation to Scientific Notation
•
•
•
•
Place the decimal point so that you have 1 digit to the left of the decimal point
___. ___ ___ ___
Count the number of places you need to move the decimal to get back to your
original number. This number will be the exponent on the 10.
If you are moving to the left, this exponent will be _____________
If you are moving to the right, this exponent will be _____________
Write in scientific notation:
8.) -873
9.) 13,000
8
10.) 0.0063
11.) -0.05
13.) 0.045 x 10-7
14.) 42.301 x 10-3
15.) 0.000375 x 108
16.) 3(1.2 x 109)
17.) 3(4.5 x 105)
18.) 0.5(1.8 x 10-9)
19.) 0.5(12.7 x 103)
20.) 5.2(4.2 x 10-2)
21.) 3.1(0.00004)
22.) 4.5(-5.3 x 108)
23.) (45,000)(123)
12.) 125.7 x 10
4
Algebra 1 Chapter 7 Notes
Order the numbers from least to greatest:
24.) 0.052 x 107, 5.12 x 105, 53.2 x 10 and 534
25.) 60.2 x 10-5, 63 x 104, 0.067 x 103, 61 x 10-2
Section 7.3
A. Writing Expressions with Exponents
Simplify. Express each in exponential form:
1.)
x x x x
4.) x2·x5·x·x3
a  a  a bbbb
2.) 3  3  3  3  3  3  3
3.)
5.) y3·y2·y4·y
6.) 3·m4·2·m2·m
What pattern did you notice with the exponents?
B. Multiplying Monomials
Multiplying Monomials
When I multiply variable expressions together, I _______________ the coefficients
AND
I ___________ the exponents of terms with the same base.
Simplify each expression. Leave answers with positive exponents.
7.) (-3m-2n5)(2mn2)
8.) (-4p3q)(-5p2q-6)
9.) (9x-3y2z)(-3x-4yz)
10.) 35 ∙ 3-2 ∙ 37
11.) 34 ∙ 22 ∙ 3-3
12.) (3c6)(c-2d8)(-2cd-5)
C. Applications to Scientific Notation
Multiplying Numbers in Scientific Notation
 Multiply Coefficients (decimal terms)
 Add Exponents
 Check to make sure that the answer is in scientific notation
13.) (2.5 x 105)(3.0 x 108)
14.) (1.6 x 10-3)(3.0 x 105)
5
Algebra 1 Chapter 7 Notes
15.) (5.4 x 10-2)(6.2 x 10-6)
16.) (9.3 x 10-4)(3.1 x 105)
17.) (8.4 x 10-7)(3.6 x 10-3)
18.) (-7.3 x 10-4)(9.1 x 108)
D. Other Applications
Complete each equation (fill-in-the-blank with a number)
 ___  6
17.) 3 ___   32  3 7
 8  820
18.) 8

19.) 3x
___ 
 3x3  9x7
20.) x4y(
)
∙ x(
)
= y2
E. Comparing Addition and Multiplication
21.) (4x2)(5x2)
vs.
4x2 + 5x2
22.) (7y3)(5y3)
23.) (14b2)(3b3)
vs.
14b2 + 3b3
24.) (7n5)(3n4)
vs.
vs.
7y3 + 5y3
7n5 + 3n4
6
Algebra 1 Chapter 7 Notes
Notes #2: Review of Sections 7.1-7.3, Section 7.4
Review of Sections 7.1-7.3
1.) Simplify: 7x-3
4.) Simplify:
2.) Simplify: 4-2
12 x 4
4 x 2
3.) Evaluate 4x-3 if x = -2
5.) Evaluate for a = 3 and b = -2 6.) Convert to scientific
notation: 304, 000
6abb0
7.) Convert to scientific
notation: 0.0872
8.) Convert to standard notation:
6.2 x 10 4
9.) Convert to scientific
notation: 47 x 108
10.) Simplify: (5x3y2)(-3x4y3)
11.) Simplify each:
12.) Multiply; leave in scientific
notation:
(-5.3 x 10-7)(8.1 x 1012)
(9m3)(5m3)
9m3 + 5m3
vs.
Section 7.4
A. Raising a Power to a Power
Explore:
1.) (x2)4
versus (x2)(x4)


2.) (3y3)2
versus
2(3y2)
Raising a Power to a Power
Raise the coefficient to the power; evaluate
_______________ the exponents on the variables
Simplify; leave in terms of positive exponents:
3.) (x3)4
4.) (m4)2(m3)3
5.) (g-2)-3(g4)5
6.) (2w-3)2
7.) (4d2)-3
9.) (p-2)5(p-10)
10.) (3.5)3(3.5)-2
8.) (5x2y4)3(2xy2)
7
Algebra 1 Chapter 7 Notes
11.) (3a2b-4)2(2ab3)3
12.) (35)2(3-2)4
13.) (x-2)(3xy2)4
Complete each equation (fill-in-the-blank with a number)
14.) (m4)___ = m12
15.) m4 ∙ m___ = m12
16.) (a2b4)___ =
1
ab
6 12
B. Applications to Scientific Notation
 Raise the coefficient to the given power
 Multiply the exponents
 Check to make sure that your answer is in scientific notation
Simplify. Write each answer in scientific notation
17.) (1.2 x 104)2
18.) (2 x 10-9)-2
19.) (-3 x 105)3
20.) (8 x 10-5)2
Simplify. Leave your answer in scientific notation.
21.) (5.6 x 104)(6.2 x 107)
22.) (3.2 x 106)2
23.) (4.3 x 108)(12.5 x 104)
24.) (3.2 x 10-5)2
25.) 4(6.3 x 107)
26.) (4.3 x 108)(12.5 x 104)
Combine like terms:
27.) 7m – 3n + 6n – 8m
28.) 3x2 – 7x + 8x2 – 2x
29.) 3ab4 – 2ab3 + ab4 + 8ab2
30.) 6j3k2 – 7j3k2 + 8jk – jk
31.) 19w8z3 – 8w7z2 + 11w8z3
32.) 10x4y3z – 2x4y3z2 + 4x4y3z
8
Algebra 1 Chapter 7 Notes
Review Topics:
1.) Multiply. Leave your answer in scientific
notation:
 6.2 x 104  3.1 x 1011 
3.) Are the lines parallel, perpendicular, or
neither?
y  3x  1
2.) Find f(-3) if f(x) = 4 – 2x2
4.) Find the x-intercept and the y-intercept of
the line:
2x – 7y = 10
3x  2 y  7
5.) Solve: 2 x  1  5
6.) Solve for x and y:
4 x  2 y  16
11x  3 y  7
7.) Graph the linear inequality:
y
10
9
3x  2 y  10
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
9
Algebra 1 Chapter 7 Notes
Notes #3: Review and Section 7.5
Review:
Combine like terms. Distribute/Multiply first if necessary.
1.) x 2  3x  2 x 2  x
2.) 2 y 2  6 y  3 y 2  4 y  9  y  5
3.) 3x(2x – 5) + 4x(8 – 3x)
4.) -7x2(6x – 2) + 3x(8 – 2x2)
6.) (4x3)2 – (5x4)(6x2)
5.) 5x(3x – 2y) – 2x(3y – 7x)
7.) (3a3)(-4a5) + (2a2)(5a4) + (3a2)4 – (2a3)2
Review Topics:
8.) Multiply; leave in scientific notation:
(8.2 x 108)(6.1 x 10-3)
9.) Evaluate; leave in scientific notation:
(4.3 x 105)2
10.) Simplify: (3a3b6)(-5a4b-9)
11.) Simplify: (3w3z8)2(-2wz2)
 6m 2 n 5 
12.) Simplify:  1 2 
 2m n 
13.) Simplify each:
3
(-4b6)(5b6)
-4b6 + 5b6
vs.
Section 7.5: Division Properties of Exponents
A. Dividing Monomials
Explore:
a6
1.) 2
a
2.)
x4 y3
x2 y
3.)
57
53
10
Algebra 1 Chapter 7 Notes
Dividing Monomials
**When I divide like numbers or variables, I ___________ their exponents.**
am
 a mn
n
a
Use this property to simplify the expressions. Leave all answers in positive exponents.
65 x 4
73 m 7
g6
4.) 6
5.)
6.)
62x
7 m5
g
 x5 y 4 z 2 
8.)  3 4 
 x y z 
c 2 d 5
7.) 9 7
cd
 4 1 
10.) 

 4 
3
4
 4 x 2 y 3  5 x 2 z 8 
9.) 
6 
8 
 5 z  2 y 
 2 m 2 
12.) 
4 
 3n 
x5 y 3 z 6
11.) 12 13 16
x y z
3
B. Fractions to Negative Powers
2
2 ___ 3___
2
Explore:    ___  ___ 
3
2
3
2
This is the same as:
2

  
3




2
Fractions to Negative Powers
**When I take a fraction to a negative power, I ___________ the fraction over, and
raise it to the ____________________ power. **
a
 
b
2
13.)  
5
2
4
14.)  
3
3
n
b
 
a
n
 4 x 
15.) 

 5 
2
 3 w 4 
16.) 
3 
 2y 
2
11
Algebra 1 Chapter 7 Notes
C. Applications to Scientific Notation
Dividing Numbers in Scientific Notation
 Divide Coefficients (decimal terms)
 Subtract Exponents
 Check to make sure that the answer is in scientific notation
17.)
4.8 x 108
2.4 x 102
18.)
4.8 x 105
6.0 x 102
19.)
1.2 x 1010
3.0 x 104
 4.2 x 10 
21.)
5 2
6.9 x 105
20.)
2.3 x 104
2.1 x 104
Review Topics:
22.) Solve for x and y:
23.) Solve for x and y:
8 x  2 y  2
4 x  5 y  15
y  5 x  1
6 x  4 y  11
24.) Graph the line: 4x – y = -8
25.) Find the equation of the line with slope
2
and passing through the point (-6, 5). Leave
3
your answer in slope intercept form.
y
10
9
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
12
Algebra 1 Chapter 7 Notes
Notes #4: Chapter 7 Review
Evaluate:
1.)
 3  4 
2
2.) -33
5.) -4-2
9.)
6.)
28m8 n3
21m 2 n7
3.) (-3)4
32 x0 y 2
61 x5 y 3
4.) -5(6x)0
7.) (-4a-5b7) (-3a4b-3)2
10.) (-2b3c5)(-5b2c)(b2c)
8.) (t6)4
11.) (-4w-3)-2(6w5)
Write in standard notation:
12.) -4.602 x 10-5
13.) 7.041 x 107
Write in scientific notation
14.) 0.00354
15.) 8,900,000
3 2
17.) (5.0 x 10 )
16.) (3 x 103)(0.2 x 10-2)
1.2 x 107
18.)
4.0 x 102
Simplify:
19.)
 32 
 2 
3 
2
2
 3x 2 y 3   4 x3 z 8 
20.) 
  8 
6
 5z   y 
1
13
Algebra 1 Chapter 7 Notes
Review Topics:
1.) Divide. Leave your answer in scientific
notation:
 3.6 x 1014 
2.) Find h(-1) if h(x) = 6x – 3
 6.0 x 10 
3
3.) Does the system have no solution, one
solution, or infinitely many solutions?
4.) Simplify:
8m3  40m2
12m4  60m3
y  2x  3
4x  2 y  6
5.) Find the equation of the line with slope ½
and passing through the point (-4, 7)
7.) Simplify: (4x – y)2
6.) Find the equation of the line passing
through the points (3, 0) and (1, 10).
8.) Find g(-1) if g(x) = 5 – 3x
14
Algebra 1 Chapter 7 Notes
9.) Find the slope and y-intercept of each line.
What can you conclude?
10.) Graph: x  4 y  7
y
5 x  3 y  12
3
3
x y
5
5
10
9
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
11.) Find the equation of the line that is
parallel to y = -2x + 1 and passing through the
point
(5, 2)
12.) Find the equation of the line that is
perpendicular to y = -2x + 1 and passing
through the point (5, 2)
15
Algebra 1 Chapter 7 Notes
Algebra 1: Chapter 7 Study Guide
For #1-8, simplify each expression:
1.) (-3)0
2.) (4)-2
3.) -2-3
5.) (7m4)0
7.) 3-2w3x-3y-2
6.) (2f)-4
Name: __________________
4.)
1
50
4ab 3
8.) 1 2
2 c
For #9-11, evaluate each expression for m = -2 and n = 3
9.) m-2n
10.) 2-3m3n-2
11.) m-n
For #12-14, write each number in scientific notation:
12.) 0.0065
13.) 130,000
14.) 0.03025
For #15-17, write each number in standard notation:
15.) 4.3 x 10-3
16.) 2.75 x 107
17.) 8 x 10-5
For #18-20, is each number written in scientific notation? If not, convert to scientific notation.
18.) 44 x 103
19.) 0.5 x 10-4
20.) 3.45 x 10-2
For #21-23, rewrite each expression using the base only once
21.) (23)(25)(2-2)
58
22.) 2
5
23.) (9.7)-8(9.7)8
16
Algebra 1 Chapter 7 Notes
For #24-35, simplify each expression. If applicable, leave your answer in scientific notation.
24.) (3bc8)(-5b2c)(2b3c-2)
25.) (8.5 x 107)(-1.2 x 10-3)
26.) (d3)2(d2)4
27.) (-5m-2n3)3
28.) (2 x 103)2
29.) (-3x2y3)2(-8xy5)
2
30.)
27m7 n5
3m9 n
 4 xy 2   9 xz 2 
31.)  3  

 3z   2 y 
33.)
3.6 x 105
1.2 x 102
34.)
 3t 1 
32.)  3 
 4r 
3.5 x 106
7 x 104
2
 5 j 2 
35.) 24 j 3k  5 
 3k 
For #36-39, complete each equation (fill in the blank).
36.) 6_  65  62
37.) 3x_  2x3  6x7
38.) ab_   a _   a5b2
39.)
8385
_
 8 
2
8 8
17
Algebra 1 Chapter 7 Notes
40.) Simplify:
 6m2  3m  8   4m2  2m  1
41.) Simplify:
3x(4 xy  5 y 2 )  7 y  2 x  8 x 2 
42.) Find the slope of the line passing through the
points: (-4, 6) and (1, -4)
43.) Find the equation of the line with slope of
1
 that passes through the point (9, -4). Leave
2
in slope-intercept form.
44.) Graph the line: 2x – y = -6
45.) Solve for x:
x 3
 5
7
y
10
9
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
46.) Solve for x:
3x  1  7
-2
-3
-4
-5
-6
-7
-8
-9
-10
47.) Solve for x and y:
3 x  y  13
x  y5
49.) Are the lines parallel, perpendicular, or
neither?
y  2x  5
2x  y  5
48.) Find g(-2) if g(x) = 5 – 3x2
50.) Find the x-intercept and the y-intercept of the
line: 3x  4 y  9
18