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Chapter 7: Exponents and Polynomials
Integer Exponents
Notes 7.1
Integers: _____________________________________________________________________________
___________________________________________
Exponents: ___________________________________________________________________________
_____________________________________________________________________________________
When you have a __________________________, you’re talking about a very small value. You are often
asked to express your answer with no negative exponents. To do this, flip the negative exponent to the
other half of the fraction and make it positive.
Express the following with only positive exponents:
a.) 𝑚−5
b.) 𝑚2 𝑛−3 𝑝−1
c.) 3−4
d.) 𝑥 −𝑚
𝑥 −3
e.) 𝑎0 𝑦5
Zero Exponents
Anything to the zero power is __________, but only what is being raised to the zero power.
Examples:
a.) 40
b.) −20
c.) (−6)0
d.) 5𝑎0 𝑏
𝑎0
e.) 𝑏3
Evaluate the following:
1.) 𝑝−3 for p = 4
2.) 8𝑎−2 𝑏0 for a = -2, and b = 6
3.) 5𝑟 −3 𝑠 −6 for r = 3 and s = 1
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Chapter 7: Exponents and Polynomials
Powers of 10 and Scientific Notation
Notes 7.2
The table below shows the relationships between several powers of ten:
Power
103
102
101
100
10−1
10−2
10−3
Value
Each time you divide by 10, the exponent decreases by 1 and the decimal point moves 1 place to the left
(negative exponents). Likewise, each time you multiply by 10, the exponent increases by 1 and the
decimal point moves 1 place to the right (positive exponents).
Powers of 10
Words
Numbers
Positive Integer Exponent:
If n is a positive integer, find the value of 10𝑛 by
starting with one and moving the decimal point n
places to the right.
Negative Integer Exponent:
If n is a negative integer, find the value of 10−𝑛 by
starting with one and moving the decimal point n
places to the left.
Scientific Notation: consists of two parts: (1 < n < 9) and 10 raised to a power.
Standard Form: Some know this form as “expanded form”.
Fill in the following:
Scientific Notation
Standard Form
1.2 x 103
50,400,000,000
8.03 x 10−7
0.0000000309
5.2 x 105
120
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Chapter 7: Exponents and Polynomials
Fill in your own examples, two with positive exponents, and two with negative exponents:
Scientific Notation
Standard Form
Comparing and ordering numbers in Scientific Notation:
Order the list of numbers from least to greatest:
1.) 1.2 x 10−1, 8.2 x 104 , 6.2 x 105 , 2.4 x 105 , 1 x 10−1, 9.9 x 10−4
2.) 5.2 x 10−3, 3 x 1014 , 4 x 10−3, 2 x 10−12, 4.5 x 1030, 4.5 x 1014
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Chapter 7: Exponents and Polynomials
Multiplication Properties of Exponents
Notes 7.3
You have seen that using exponents are helpful when writing very small and very large numbers. To
perform operations on these numbers, you can use properties of exponents. You can also use these
properties to simplify expressions.
When an expression is fully simplified, follow this checklist:
 There are no negative exponents
 The same base doesn’t occur more than once in the product or quotient
 No powers, products or quotients are raised to powers
 Numerical coefficients in both numerator and denominator have no factors in common.
When multiplying:
Coefficients:
Base:
Exponents:
1.) 24 • 23
2.) 𝑦 −5 • 𝑦 3 • 𝑥 8 𝑦 −4 • 𝑦 2
3.) 𝑚 • 𝑛−3 • 𝑚4
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Chapter 7: Exponents and Polynomials
When raising a power to a power:
Coefficients:
Base:
Exponents:
4.) (74 )3
5.) (36 )0
6.) (𝑥 2 )−4 𝑥 5
8.) −(3𝑥 3 )2
9.) (𝑥 −2 𝑦 0 )3
When finding the powers of products…
Simplify:
7.) (−3𝑥 5 )4
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Chapter 7: Exponents and Polynomials
Simplify the following powers of products:
10.) (5𝑥 6 𝑦 2 )( 2𝑥 2 𝑦 4 )3
11.) (4𝑝)5
12.) (𝑥 2 𝑦 3 )4 ( 𝑥 2 𝑦 4 )−4
Applying:
13.) Light from the sun travels at about 1.86 x 105 miles per second. It takes about 500 seconds for the
light to reach Earth. Find the approximate distance from the Sun to Earth. Write your answer in
scientific notation.
14.) Light travels at 1.86 x 105 miles per second. Find the approximate distance that light travels in one
hour. Write your answer in scientific notation.
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Chapter 7: Exponents and Polynomials
Division Properties of Exponents
Notes 7.4
The quotient of powers with the same base can be found by writing the powers in factored form and
dividing out common factors. Remember in the last section when we multiply with the same base, you
add the exponents. So keeping that in mind, when we divide, we will need to subtract the exponents.
*HOWEVER, it is crucial to keep in mind where you start because that’s where you end. If you start your
subtraction in the numerator, your answer will go in the numerator. If you start your subtracting in the
denominator, your answer goes in the denominator. Remember to be fully simplified; you need to have
only positive exponents.
When dividing with exponents:
Coefficients:
Base:
Exponents:
Simplify the following:
1.)
38
32
2.)
𝑎5 𝑏9
(𝑎𝑏)4
𝑚5 𝑛4
(𝑚5 )2 𝑛
3.)
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Chapter 7: Exponents and Polynomials
Simplify the following:
3 3
4
4.) ( )
𝑎 −𝑛
7.) (𝑏 )
3
2𝑥 3
)
𝑦𝑧
5.) (
2 −3
8.) (5)
3
𝑎3 𝑏
)
𝑎2 𝑏2
6.) (
3 −1
9.) (4)
2𝑥 −2
(3𝑦)
Exception: Only with scientific notation can the exponent be negative in your answer.
10.) (2 x 108 ) ÷ (8 x 105 )
11.) (3.3 x 106 ) ÷ (3 x 108 )
12.) In the year 2000, the United States public debt was about 5.6 x 1012 dollars. The population of the
United States in that year was about 2.8 x 108 people. What was the average debt per person? Give
your answer in standard form.
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Chapter 7: Exponents and Polynomials
Polynomials
Notes 7.5
Monomial: ___________________________________________________________________________
_____________________________________________________________________________________
Degree of Monomial: ___________________________________________________________________
_____________________________________________________________________________________
State the degree of the following Monomials:
1.) 5
2.) -5𝑥 1
3.) 6𝑎5 𝑏 2
4.) -2𝑎2 𝑏
5.) 4𝑥 0
Polynomial: ___________________________________________________________________________
_____________________________________________________________________________________
Degree of a Polynomial: _________________________________________________________________
_____________________________________________________________________________________
Find the degree of the following:
6.) 4x - 18𝑥 5
7.) .5𝑥 2 𝑦 + .25xy + .75
Standard form of a Polynomial: __________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
Leading Coefficient: ____________________________________________________________________
_____________________________________________________________________________________
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Chapter 7: Exponents and Polynomials
Names for Polynomials: First name, Last name = Degree name, Term name
Degree
Name
Terms
0
1
1
2
2
3
3
4 or more
Name
4
5
6 or more
Name the following:
8.) 𝑥 3 + 𝑥 2 - x + 2
9.) 6
10.) -3𝑦 8 + 18𝑥 2 𝑦 3 + 14y
11.) 4𝑥 2 y + 8x + 2𝑥 2 y𝑧 2 - 5𝑥 3 𝑧 0 – 7
12.) A firework is launched from a platform 6 feet above the ground at a speed of 200 feet per second.
The firework has a five second fuse. The height of the firework in feet is given by the polynomial:
-16𝑡 2 + 200t + 6 where t is the time in seconds. How high will the firework be when it explodes?
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Chapter 7: Exponents and Polynomials
Adding and Subtracting Polynomials
Notes 7.6
Like terms must have two things in common:
1.) Same __________________
2.) Same __________________
Rules for adding and subtracting with exponents:
1.) The base stays the same
2.) The exponents stay the same
3.) Apply the math to the coefficients
*Always express your answer in standard form!
Two methods:
Vertical method: Line up the problem so that the like terms are on top of each other and the
operation can be performed vertically. Remember, with subtraction, first do keep change
change.
1.) (15𝑚3 + 6𝑚2 ) + 2𝑚3
2.) (-4𝑔3 - 13𝑔2 + 2g) – (-5𝑔2 + 8g – 9)
Horizontal method: Line up the problem in one horizontal row by grouping the like terms
together to simplify.
3.) (𝑎4 - 2a) – (3𝑎4 - 3a + 1)
4.) (3𝑥 2 - 2x + 8) + (x - 𝑥 2 - 4)
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