Math Integrated Algebra 1A
Unit 9: Exponents and Polynomials
Property
Example(s)
Rule
Zero Exponents
Negative Exponents
Multiplying Powers
(Same Base)
Dividing Powers
(Same Base)
Power to Power
Powers of Products
Adding and Subtracting
Polynomials
Multiplying Polynomials
1
Math Integrated Algebra 1A
Day 1 - Zero and Negative Exponents
Unit 9: Exponents and Polynomials
Vocabulary
2
Simplifying Powers
To simplify a base in the numerator that has a negative exponent, move the base
to the denominator and write a ___________________________ exponent.
To simplify a base in the denominator that has a negative exponent, move the base
to the numerator and write a _____________________________ exponent.
Write T for true or F for false.
1) ______
9-2 =
2) ______ 9-1 = 
1
81
3) _____
1
9
4) ______ 9-3 =
(-9)0 = -1
1
27
What is the simplified form of each expression (using only positive exponents)?
2)
2
a 3
3) 4c-3b
4)
n 5
m2
1
5) 3
n
 2
6)  
 9
1)
x-9
2
3
Evaluating an Exponential Expression
What is the value of n-4w0 for n = -2 and w = 5?
Evaluate each expression for a = -4, b = 3, and c = 2.
1) 3a-1
2) b-3
3) 4a2b-2c3
4) 9a0c4
5) -a-2
6) (-c)-2
Mixed Practice:
Simplify each expression.
1. 370
3.
5
52
5. (5)2
2. 34
4.
3
61
6. 1121
4
7) The number of people who vote early doubles every week leading up to an election.
This week 1200 people voted early. The expression 1200 • 2w models the number of
people who will vote early w weeks after this week. Evaluate the expression for
w = –3. Describe what the value of the expression represents in the situation.
8) The population of a suburb is 4000 people. The population of the suburb is expected to
double each decade. The expression 4000 • 2d models the population of the suburb
after each decade d. Evaluate the expression for d = 2. Describe what the value of
the expression represents in this situation.
5
Day 1 – Homework
Simplify each expression.
1. 130
3.
3
3
4
5. –(7)–2
7. –60
9.
2. 5–3
4.
2
4 4
6. 46–1
8. –(12x)–2
1
80
10. 6bc0
6
For 11–16, evaluate each expression for x = -2, y = 4, and
z = 2.
11) 4x
1
13) 2xy - 2z
15) x
2
12) z 3
2
14) 6x 3z
0
16) (y) 3
7
Math Integrated Algebra 1A
Day 2 – Multiplying Monomials
Unit 9: Exponents and Polynomials
Warm Up
Circle the base in each expression
1.
54
2. (-3)8
3. n-5
Error Analysis A student writes an expression with exponent 4 and base 3. Is the
student’s expression 43 correct? Explain.
When multiplying powers with the same base, you add the exponents.
This is true for numerical and algebraic expressions.
What is each expression written as a single power?
a. 34 • 32 • 33
All three powers have the same base, so this expression can be written as a single
power by adding the exponents.
34 • 32 • 33  34 + 2 + 3
exponents.
 39
All powers have the same base. Add the
Simplify the exponent.
Even when an expression contains negative or rational (fractions) exponents, the
exponents can be added when the bases are the same in a product of powers.
b. 113 • 114 • 115
113 • 114 • 115  113 + 4 + (5)
All powers have the same base. Add the
exponents.
 114
Simplify the exponent.
8
1
3
1
3
c. 2 5  2 5
1
25  25 = 25

3
5
All powers have the same bases. Add the exponents.
4
Simplify the exponent.
 25
What is each expression written using each base only once?
a) 83
86
_______
b) (0.5)-3 (0.5)-8 _______
c) 9-3
92
96_______
What do we do if there is a coefficient in front of the base?????
What is the simplified form of 5x4
x9
3x?
Simplify each expression.
1.
z 8z
5
4. (13x–8)(3x10)
7. mn2 • m2n–4 • mn –1
2. –4k– 3 • 6k 4
3. (–5b3)(–3b6)
5. (–2h5)(4h–3)
6. –8n • 11n 9
8. (6a 3b –2)( –4ab –8)
9. (12mn)(–m3n-2p5)(2m)
9
Scientific Notation
... is a way to express very small or very large numbers.
... is most often used in "scientific" calculations where the analysis must be very
precise.
... consists of two parts*:
(1)
a number between 1 and 10, such that
and
(2)
a power of 10.
* a large or small number may be written as any power of 10; however, CORRECT
scientific notation must satisfy the above criteria.
3.2 x 1013
is correct scientific notation
23.6 x 10-8
is not correct scientific notation
Remember that the
first number MUST BE
greater than or equal
to one and less than 10.
To multiply two numbers expressed in scientific notation,
simply multiply the numbers out front and add the exponents.
Generically speaking, this process is expressed as:
(n x 10a) • (m x 10b) = (n · m) x 10a+b
Example 1: (5.1 x 104) • (2.5 x 103) = 12.75 x 107 Oops!!
This new answer is no longer in proper scientific notation.
Proper scientific notation is 1.275 x 108
10
What is the simplified form of (1.8 × 1011)(2.7 × 108)?
Write the answer in scientific notation.
Use the Associative and Commutative Properties of Multiplication to regroup and
reorder the factors so that the powers of 10 are grouped together and numbers
that are not powers of 10 are grouped separately from the powers of 10.
(1.8 × 1011)(2.7 × 108)  (1.8 • 2.7)(1011 • 108) Associative and Commutative Prop. of Mult.
 (4.86)(1011 + 8 )
Multiply the numbers in the first set of
parentheses. Add the exponents for the powers
of 10.
 4.86 × 1019
Simplify the exponent.
Simplify each expression. Write each answer in scientific notation.
1.
(7 × 1017)(8 × 10–28)
2. (4 × 10–11)(0.8 × 107)
3. (0.9 × 1015)(0.1 × 10–6)
4. (0.8 × 105)(0.6 × 10–17) 5. (0.5 × 103)(0.6 × 100) 6. (0.2 × 1011)(0.4 × 10–14)
11
7.
The diameter of the moon is approximately 3.5 × 103 kilometers.
a. The diameter of Earth is approximately 3.7 times the diameter of the moon.
Determine the diameter of Earth. Write your answer in scientific notation.
b. The distance from the center of Earth to the center of the moon is
approximately 30 times the diameter of Earth. Determine the distance
from the center of Earth to the center of the moon. Write your answer in
scientific notation.
12
Day 2 – Homework
Simplify each expression.
1.
a2a3
4.
(8p
7.
h
5
5
)(6p4)
•h2•h
10
(y3z13)(y2z6)
13.
m
•m
3
3n3n5
3.
8k3 • 3k6
5.
21d 2 2 d 4
6.
(6.1m4)(3m2)
8.
(9q –8)(6q 11)
9.
(16r
 3 x 12  5w8  4 x 13 
 



11. 
12.
(15fg2)(f 3g3)( 8f 1g6)
6j 3k • 7jk5
15.
2uvw 1 • 3u2v2w
1
10.
6
2.
• n 2
14.
1
7)(
2r)
Simplify each expression. Write each answer in scientific notation.
16.
(4 × 103)(2 × 105)
17.
(1 × 104)(6 × 103)
18.
(7 × 102) • 105
19.
(8 × 109)(3 × 105)
20.
(2 × 105)(5 × 106)
21.
(7 × 108)(3 × 106)
13
Math Integrated Algebra 1A
Unit 9: Exponents and Polynomials
Day 3 – Power to a Power Rule
Extra Practice: Powers of Products and Quotients Worksheet
Warm Up
Simplify each power raised to a power.
1) (42)3 _______ 2) (54)2 _______ 3) (w5)2 _______ 4) (m1/3)5 _______
Simplify each product raised to a power.
1) (7m9)3 _________
2) (2z)-4 _________
3) (4a1/2)2 _________
14
Scientific Notation
Simplify. Write each answer in scientific notation.
1. (5 × 107)2
2. (2 × 104)6
3. (9 × 1012)2
4. (3 × 108)3
5. (3.6 × 105)2
6. (1.7 × 108)3
Mixed Practice
Simplify each expression.
1. (z5)3
2. (m4)10
5. (x7)2
 1
6.  r 4 
 
3
1
9. (m2 ) 2 n 7
1
4
3. (v 7 ) 2
4. (k 3 )3
7. b(b8)3
8. h 2(h 7)
6
10. (x6)2(y3)0
11. (g5)5(g6)2
0
1
12. (v 2 )3 ( w4 ) 3
15
Day 3 – Homework
Simplify each expression.
1. (v4)2
2. (a5)7
4. h(h–3)–10
4 6
5 –2
7. (a ) (b )
10. (5k9)–3
5.
2
3 3
x 
2 8
3. (n4)–6
y3
0 4
8. (m ) (n )
11. (3s4)3s2
6. (3q)5
 1
9.  2t 2 


4
12. (bc3)1/5
Simplify. Write each answer in scientific notation.
13. (4 × 103)2
14. (7 × 108)4
16. (5 × 10–7)3
17. (2.5 × 1010)2
15. (1.4 × 10–8)3
16
Math Integrated Algebra 1A
Unit 9: Exponents and Polynomials
Day 4 – Dividing Monomials
Extra Practice: Exponents & Division Worksheet
Warm Up
Simplify each expression.
5
6
1.
5
52
2.
5
5
52
3.
x8
3
4.
m 3
m 5
x8
17
Dividing Numbers in Scientific Notation
Simplify each quotient. Write each answer in scientific notation.
1.
3.6  107
2.
1.5 0
4.5  10 6
5  10 2
3. Population density describes the number of people per unit area. During
one year, Honduras had a population of 7.33 x 106 people. The area of Honduras is
4.33 x 104mi2. What was the population density of Honduras that year?
Hint: Unit Area = Population/Area (Final Answer is People Per Mile)
Simplify each expression:
1.
 3
 
 10 
4
6
2.  
2
2
 42 
3.  3 
2 
 64 
4. Error Analysis A student simplifies the expression  2 
3 
3
3
 64  
4 2 
2
 2  = 6  3  = 2
3 
3
3
as follows:
   64.
3
What mistake did the student make in simplifying the expression? What is the
correct simplification?
18
Day 4 – Homework
Simplify each expression.
1.
3.
5
3
3
2
y
7
y
4
3 3 3 3 3
3
3 3

2.
6
7
6
3
4
4. m
m
3
5.
6 9
x y
6.
2 5
x y
2
7.  
7
21m 2
1
3m 2
4
 5x 
9.  
3y 
3
8.  
2
2
 34 
2m 
11. 
 5p 


3
3
10.
 4
 3x 
 2 y3 


2
12.
 3
 xy 
 x3 y 


0
Simplify each quotient. Write each answer in scientific notation.
13.
6  10
7
3  10
5
14.
2.4  10
8.2  10
3
2
19
Math Integrated Algebra 1A
Unit 9: Exponents and Polynomials
Day 5 –Adding and Subtracting Polynomials
CW/HW: Algebra With Pizzazz! Pgs 61 & 62
Warm Up
20
The "Standard Form" for writing down a polynomial is to put the terms with the
highest degree (largest exponents) first.
Polynomials are in simplest form when they contain no like terms.
x2 + 2x + 1 + 3x² - 4x when simplified becomes 4x2 - 2x + 1
Polynomials are generally written in descending order.
Descending: 4x2 - 2x + 1 exponents of variables decrease from left to right.
Mixed Practice.
Simplify.
1.
11n – 4
– (5n + 2)
2.
4. (28e3 + 3e2) + (19e3 + e2)
7x4 + 9
– (8x4 + 2)
3.
3d2 + 8d – 2
– (2d2 – 7d + 6)
5. (–12h4 + h) – (–6h4 + 3h2 – 4h)
21
6. A small town wants to compare the number of students enrolled in public
and private schools. The polynomials below show the enrollment for
each:
Public School: –19c2 + 980c + 48,989
Private School: 40c + 4046
Write a polynomial for how many more students are enrolled in public
school than private school.
Simplify. Write each answer in standard form .
7. (3a2 + a + 5) – (2a – 5)
8. (6d – 10d3 + 3d2) – (5d3 + 3d – 4)
9. (–4s3 + 2s – 3) + (–2s2 + s + 7)
10. (8p3 – 6p + 2p2) + (9p2 – 5p – 11)
22
Math Integrated Algebra 1A
Unit 9: Exponents and Polynomials
Day 6 – Multiplying a Monomial and a Polynomial
Monomials can be multiplied times other monomials, times binomials, times
trinomials, and times polynomials in general.
23
Simplify each product.
1. 2x(x + 8)
2. (n + 7)5n
3. 6h2(7 + h)
4. –b2(b – 10)
5. –3c(8 + 2c – c3)
6. y(2y2 – 3y + 6)
7. 4t(t2 – 6t + 2)
8. –m(4m3 – 8m2 + m)
9. 7j(–2j2 – 8j – 3)
10. –t2(2t4 + 4t – 8)
11. 2k(–3k3 + k2 – 10)
12. 8a2(–a7 + 7a – 7)
13. 4v3(2v2 – 3v + 5)
14. 5d(–d3 + 2d2 – 3d)
15. 11w(w2 + 2w + 6)
24
Simplify. Write in standard form.
16. –3x(4x2 – 6x + 12)
17. –7y2(–4y3 + 6y)
18. 9a (–3a2 + a – 5)
19. p(p + 4) – 2p(p – 8)
20. t(t + 4) + t(4t2 – 2)
21. 6c(2c2 – 4) – c(8c)
22. –5m(2m3 – 7m2 + m) 23. 2q(q + 1) – q(q – 1)
24. –n2(–6n2 + 2n)
25
Day 6 – Homework
Simplify each product.
1. 3w(w + 2)
2. (z + 5)2z
3. 3m2(4 + m)
4. 2p(p2  6p + 1)
5. y(5y3  3y2 + 2y)
6. 3a(3a2 + 2a  7)
7. 6x3(3x2  x + 10)
8. 4h(h3  8h2 + 2h)
9. 4n(n2 + 5n + 6)
26
Math Integrated Algebra 1A
Day 7 – Multiplying Binomials (Box & Foil)
Unit 9: Exponents and Polynomials
How do we multiply a BINOMIAL by a BINOMIAL?????
METHOD 1:
“Double Distribute”
(x + 3)(x - 2) = x2 + x – 6
1. Distribute the first binomial to each term in the second
2. Distribute
3. Simplify by combining like terms.
(x + 3)(x - 2) = x(x + 3) - 2 (x + 3)
= x(x) + x(+3) - 2(x) -2(+3)
= x2 + 3x - 2x - 6
= x2 + x - 6
Step 1
Step 2
Step 3
IF MULTIPLYING TWO BINOMIALS, DOUBLE DISTRIBUTION IS ALSO
KNOWN AS “FOIL”
27
PRACTICE:
1.
(a - 1)(a + 4)
2. (x - 6)(3x - 2)
3.
(2x + 5)(x -2)
4. (3 - x)(8 + x)
5.
(x + 7)(x + 9)
6. (x + 2)2
7. (a - 2)(a + 4)
8. (x + 1)(3x - 2)
9.
(x + 5)(x -2)
10.
11.
(x - 7)(x + 9)
(3 - x)(4 + x)
12. (x - 2)2
28
METHOD 2: “BOX METHOD”
29
Day 7 – Homework
Simplify each product using the Distributive Property.
1. (b  2)(b + 1)
2. (x + 6)(x + 5)
3. (3n + 1)(n  8)
4. (2t  7)(t  5)
5. (y + 3)(y + 7)
6. (b  6)(b + 3)
Simplify each product using the Box Method.
7. (x + 1)(x  11)
8. (h  2)(3h + 5)
9. (8w  3)(4w  7)
30
Math Integrated Algebra 1A
Day 8 – Perimeter and Area Problems
Unit 9: Exponents and Polynomials
Find the area of the shaded region in the simplest form.
(BIG SHAPE) – (LITTLE SHAPE “HOLE”) = SHADED REGION
Warm Up:
1) A square of side length 8 has a right triangle with a base of 4 and a height of 3
cut out of it.
a) Find the perimeter of the square ________________
b) Find the perimeter of the triangle _______________
c) Find the area of the square __________________________________
d) Find the area of the triangle __________________________________
e) Find the area of the shaded region _____________________________
2) A rectangle with width of 7 and length of 9 has a square of side length 5 cut out
of it.
a) Find the perimeter of the square ________
b) Find the perimeter of the rectangle______
c) Find the area of the square ___________________
d) Find the area of the rectangle ____________________
e) Find the area of the shaded region ____________________
31
3) Given a rectangle whose sides are 3x+7 and 4x-9, find the polynomial that
represents the:
a) perimeter
b) area
4) The sophomore class is working on a float for the homecoming parade. They
need to put a fringe around the perimeter of the rectangular trailer. The
length is 6 feet longer than the width. If 52 feet of trim is used, what is
the length and width of the trailer?
5) What expression represents the perimeter, P, of the triangle shown?
32
For 6-8, find the area of the shaded region:
6)
11y
6y
6y
11y
7)
3x
4x
3x
5x - 2
8)
8 -2t
3-t
t
3t
9) You are painting a rectangular wall with length 5x2 ft and width 12x ft. There is
a rectangular door that measures x ft by 2x ft that will not be painted. What is
the area of the wall that is to be painted?
33
Day 8 – Homework
1) The recreation field at a middle school is shaped like a rectangle with the length
of 25x yards and a width of 20x − 4 yards. Write a polynomial for the
perimeter of the field. Then calculate the perimeter if x = 4.
2) A bedroom has a length of x − 5 feet and a width of x + 4 feet. Write a
polynomial to express the area of the bedroom. Then calculate the area if x = 9
34
Math Integrated Algebra 1A
Day 9 – Review Grid Activity
HW: Quiz 2 Review Worksheet
Unit 9: Exponents and Polynomials
Quiz 2 Review Worksheet
For #1-3 state whether each expression is a monomial, binomial or trinomial
1. 9x – 3
_______________
2. 2a + 4a – 5 _______________
3. 10y
_______________
Write the algebraic expression in simplest form, circle final answer:
4. 8x + 5x – 2x
5. 3r – (r + 3)
6. (4x – 7) + (9x + 2)
7. (5w + 3) – (6x – 5)
35
8. (x + 2)(x – 3)
9. (x + 5)2
10. (r4) (r3)
11. (a2b)3
12. (-8w5) (9w)
13. –ab(a – b)
14. 10d(2a – 3c + 4b)
15. 3xy(x2 + xy - y)
36
Use the drawing below to answer #16-19:
4x
2x
4x - 2
6x + 3
16. Find the area of the larger rectangle: __________________
17. Find the area of the smaller rectangle: _________________
18. Find the area of the shaded region: ___________________
19. Find the perimeter of the larger rectangle: ________________
37
Math Integrated Algebra 1A
Day 10 - Multiplying Polynomial by Binomial
HW: Algebra With Pizzazz! Pg 69
Unit 9: Exponents and Polynomials
WARM-UP: Multiply
1.
(x - 4)(x - 5)
3.
(3 - x)(4 + x)
2.
(2x + 3)(x + 1)
4.
(x - 5)2
To Multiply a Binomial and a Trinomial
METHOD 1: "Distribute" (like FOILing, but without the acronym)
38
METHOD 2: "Box Method"
PRACTICE PROBLEMS
1.
(x + 2)(x2 - x + 3)
4.
(x - 8)(x2 + 2x - 1)
2.
(x - 5)(2x2 - 3x + 3)
5.
(x - 5)(-x2 - 3x - 5)
3.
(2x + 1)(x2 + 2x + 7)
6.
(2x + 1)(2x2 + 6x - 4)
39
Math Integrated Algebra 1A
Day 11 - Dividing a Polynomial by a Monomial
HW: Algebra With Pizzazz! Pg 77
TRY:
1.
(8a5 - 6a4+ 2a2) ÷ 2a2
Unit 9: Exponents and Polynomials
2.
(15cd + 11c) ÷ 5c
40
Integrated Algebra 1A
Unit 9: Exponents and Polynomials
Day 12 – Perfect Squares & Perfect Cubes
HW: Algebra With Pizzazz! (Pg 158)
List all the Perfect Squares 1 through 15
List all the Perfect Cubes 1 through 4
41
Day 13 – Test Review Day 1
HW: Test Review Day 2
Multiplying Monomials:
xa xb = ________________
(xa)b = ______________
Dividing Powers that have the Same Base:
keep the base and ______________________ the exponents.
xa


xb = ________________
Zero Exponent:
Ex. 56


52 ______________
When x  0, x0 =1.
Negative Exponents: (reciprocal)
Ex.
When x  0, x-n =
Ex. 5-3 = _________________
70 = ______________
1
xn
Ex.
1
= _________
32
Dividing Monomial by Monomial:
1. Divide the numerical coefficients
2. Divide powers of the same base by substracting exponents
3. Simplify
Ex.
24x 5 y 4 z
=
3x 2 yz
Ex.
35a 3bc
=
7ab 3
Dividing a Polynomial by a Monomial:
Divide each term of the polynomial by the monomial.
Ex.
42x 3 y 4  18x 2 y2  6xy
=
6xy
Ex.
2a 2  3a  1
=
a
42
Practice
For 1-8, simplify each expression:
1. 24x3y4z2 + 12x2y3z – 6xy2
6xy2
2. 3y2 – 2y + y2 – 8y – 2
3. 5t – (4 – 8t)
4. 8mg(-3g)
5. 3x2(4x2 + 2x – 1)
6. (4x + 3)(2x – 1)
7. (2a – 5)2
8. 40b3x6  -8b2x
43
For 9-10, write each expression in an equivalent form, using positive exponents only:
9. k-6
2
10. ( )-2
5
For 11-12, use the laws of exponents to perform the operations and simplify:
11. 3-5  34
12. (10-2)-1
13. If one side of a square is x + 5, write the polynomial that represents the perimeter of the
square and the polynomial that represents the area of the square.
44
Day 13 HW - Test Review Day 2
Simplify each expression:
1.
(x – 2)(x – 4)
2. (2x + y)(2y + x)
3.
(3 + 2x)(2 + 3x + 2x2)
4. x3 ÷ x2
5.
x0
6. x15c ÷ x5c
7.
37
33  3
8. 10
9.
4 -3 ÷ 4
11.
Transform into an expression with a positive exponent: x
-2
-4
∙ 10
3
10. (3-2)-4
-8
45
Simplify. Write each answer in scientific notation.
12. (8.18 × 10−6)(1.15 ×10−5)
13. (5.8 × 10−6)(2 × 104)
14. (0.8 ×104)(1.28 ×106)
15.
7.8 ×104
8 ×101
16.
2 × 103
4 × 105
17.
.8 × 102
2 ×10−3
46
Without using your calculator!
Write out the first 15 Perfect Squares...and their Square Roots….
Integers
1 x 1 or (12)
2 x 2 or (22)
3 x 3 or (32)
4 x 4 or (42)
5 x 5 or (52)
6 x 6 or (62)
7 x 7 or (72)
8 x 8 or (82)
9 x 9 or (92)
10 x 10 or (102)
11 x 11 or (112)
12 x 12 or (122)
13 x 13 or (132)
14 x 14 or (142)
15 x 15 or (152)
Perfect Square
1
4
Square Root
1
2
Write out the first 4 Perfect Cubes...and their Cube Roots….
Integers
1 x 1 x 1 (or 13)
2 x 2 x 2 (or 23)
3 x 3 x 3 (or 33)
4 x 4 x 4 (or 43)
Perfect Cube
1
8
Cube Root
1
2
47