Algebra 1A UNIT 7: Exponents
LESSON 7-1: Zero and Negative Exponents
LESSON 7-2: Multiplying Powers with the Same Base
LESSON 7-3: More Multiplication Properties of Exponents
LESSON 7-4: Division Properties of Exponents
LESSON 7-1: Zero and Negative Exponents
OBJECTIVE: To simplify expressions involving zero and negative exponents.
RECALL:
You can use EXPONENTS to show _______________ multiplication.
A POWER has two parts, a BASE and an EXPONENT.
An __________ tells you how many times a number, or _______ is used as a factor, or repeated.
5  5  5  5  125
3
READ as “Five to the third” or “Five cubed”
PROPERTIES OF EXPONENTS:
ZERO EXPONENTS:
For every nonzero number a, a0 = 1
Examples: 40 = 1
(-3)0 = 1
(5.14)0 = 1
NEGATIVE EXPONENTS:
1
an
1


 52
For every nonzero number a and integer n, a  n 
Examples: 7 3 
1

73
 52
Why can’t zero be used as a base for negative exponents?
What is 0 0 ?
Examples 1a – 1h: What is the simplest form of each expression?
a.) 9 2
b.)  3.6 
0
c.) 4 3
d.)  5
0
e.) 3 2
f.) 6 1
h.)  4
2
g.)  4 2
Examples 2a – 2h: What is the simplest form of each expression?
1
a.) 5a 3 b 2
b.) x 9
c.) 5
x
e.)
1
n 3
f.)
2
a 3
g.)
Example 3: What is the value of 3s3t-2 for s = 2 and t = -3?
Method 1: Simplify First
d.) 4c 3 b
n 5
m2
h.)
x 0 y 3
z 2
Method 2: Substitute First
Example 4: What is the value of 2a-2b4 for a = -2 and b = 1?
Examples 5a – 5d: What is the value of each expression in parts a-d for n = -2 and w = 5?
a.) n-4w0
b.)
n 1
w2
c.)
n0
w6
d.)
1
nw 1
Example 6: A population of birds double every 8 years. The expression 6400 ∙ 2t models a
population of 6400 birds after t periods of 8 years. Evaluate the expression for t = 0 and t = -2.
Describe what each value of the expression represents in the situation.
Example 7: A population of insects doubles every week. The number of insects is modeled by the
expression 5400 ∙ 3w, where w is the number of weeks after the population was measured. Evaluate
the expression for w = 0, w = 0 and w = -2. Describe what each value of the expression represents in
the situation.
LESSON 7-2: Multiplying Powers with the Same Base
OBJECTIVE: To multiply powers with the same base.
BELL RINGER: What is the value of -6a-3b2 for a = -2 and b = 4?
NOTES
How would you write 34 ∙ 32
EXPONENT PROPERTY:
MULTIPLYING POWERS with the SAME BASE:
To multiply powers with the same base, add the exponents.
am ∙ an = am + n, where a ≠ 0 and m and n are rational numbers.
1
3
1
3
Examples: 4  4  4
1 1

3 3
4
2
3
b 7  b 4   b 7  4   b 3
Examples1a – 1f: What is each expression written using each base only once?
a.) 124 ∙ 123
b.) (-5)-2 ∙ (-5)7
c.) 86 ∙ 83
d.) (0.5)-3 ∙ (0.5)-8
e.) 92 ∙ 9-3 ∙ 96
f.) (-3)-2 ∙ (-3)10
Examples 2a – 2f: What is the simplified form of each expression?
a.) 4z5 ∙ 9z-12
b.) 3a2 ∙ 9b4 ∙ 2a
c.) 5x4 ∙ x9 ∙ 3x
d.) -4c3 ∙ 7d2 ∙ 2c-2
e.) j2 ∙ k-2 ∙ 12j
f.) 6b3 ∙ 3b-8
Example 3: Explain how to simplify the expression xa ∙ xb ∙ xc.
Example 4: The speed of light is about 3 × 108 meters per second. How far would a beam of light
travel in 8.64 × 104 seconds (1 day)?
Example 5: At 20⁰C, one cubic meter of water has a mass of about 9.98 × 105 g. Each gram of water
contains 3.34 × 1022 molecules of water. About how many molecules of water does a droplet of water
contain if its volume is 1.13 × 10-7 m3?
Example 6: About how many molecules of water are in a swimming pool that holds 200 m 3 of water?
Write your answer in scientific notation.
Exponents can also be expressed as fractions. Fractional exponents are called rational exponents.
Examples 7a – 7h: Simplify each expression.
1
1
1
1
a.) 81 4
b.) 16 4
c.) 27 3
d.) 64 2
e.) 64
3
2
f.) 25
3
2
g.) 27
2
3
3
h.) 16 4
Examples 8a – 8d: Simplify each expression.
1
1
 23
 1

4  3

a.)  2a  3b  a  5b 2 



 23 52  94 109 
b.)  b  c  b  c 



2
1
 2
 1

c.)  3 j 3  7 m 4  3 j 6  7m 3 



1
 3

d.)  2c 5  2c 5 


LESSON 7-3: More Multiplying Properties of Exponents
OBJECTIVE: To raise a power to a power.
To raise a product to a power.
1
3
 13
 1

2  3

BELL RINGER: Simplify  2a  3b  a  3b 4  .



NOTES
How would you write x 5  ?
2
EXPONENT PROPERTY:
RAISING A POWER TO A POWER:
To raise a power to a power, multiply the exponents.
a 
m n
 a mn , where a ≠ 0 and m and n are rational numbers.
b 
3 5
Examples: 54  52  542  58
3
 b35  b15
3
13
3

 12  5
 x   x 2 5  x 10
 
 
9
3
3
 32 
a   a2  a3
 
 
Examples 1a – 1f: Simplify each expression?
1

 2 2
b.)  n 3 
 
d.) b3 
 23  5
e.)  x 
 
a.) n
4 7
c.)  p 5 
4
3
5
3
 12  4
f.)  x 
 
Example 2: Is a m   a mn true for all integers m and n? Explain.
n
Examples 3a – 3d: Simplify each expression.
 52 
a.) y  y 
 
2
b.) x 2 x 6 
4
3
 5
c.) w  w 3 
 
3
d.) h  2 h 4 
3
2
4
 1
How would you write  3m 2  ?


EXPONENT PROPERTY:
RAISING A PRODUCT TO A POWER:
To raise a product to a power, raise each factor to the power and multiply.
ab n  a n  b n , where a ≠ 0, b ≠ 0, and n are rational numbers.
Examples:
3x 4  34 x 4  81x 4
3
3
3
 32 
 4b   4 2  b 2  8b 2




Example 4: Write an expression for the area of a square with side length 5x3.
Examples 5a – 5c: Simplify each expression.
a.) 7m9 
c.) 3g 4 
b.) 2 z 
2
4
3
Examples 6a – 6d: What is the simplified form of each expression?
 1
a.)  b 3 
 
6
1
 

 2ab 2 




10
 1
c.)  n 2 
 
4
2
 

 4mn 3 




b.) x 2  3xy5 
2
3
4
d.) 6ab  5a 3 
3
2
4
Example 7: The formula for the volume of a sphere is v  r 3 , where r is the radius. What is the
3
2
volume of a sphere with a radius of 10 millimeters? Give your answer in terms of π.
1 2
mv gives the kinetic energy, in joules, of an object with a mass of m kg
2
traveling at a speed of v meters per second. What is the kinetic energy of an experimental unmanned
jet with a mass of 1.3 × 103 kg traveling at a speed of about 3.1 × 103 m/s?
Example 8: The expression
Example 9: What is the kinetic energy of an aircraft with mass of 2.5 × 105 kg traveling at a speed of
about 3 × 102 m/s?
LESSON 7-4: Division Properties of Exponents
OBJECTIVE: To divide powers with the same base.
To raise a quotient to a power.
BELL RINGER: Simplify z  2  7 yz 3  .
3
2
NOTES
How would you write
45
?
43
EXPONENT PROPERTY:
Dividing POWERS with the SAME BASE:
To divide powers with the same base, subtract the exponents.
am
 a m  n , where a ≠ 0 and m and n are rational numbers.
n
a
6
4
s
x
1
 x 47  x 3  3
7
x
x
2
Examples: 2  26  2  24
2
s
3
4
1
2
3 1

2
 s4
1
 s4
Examples 1a – 1f: Simplify each expression.
5
2
a.)
2
x
x2
b.)
4
mn
m5 n3
a 3b7
e.) 5 2
ab
k 6 j2
d.)
kj5
c.)
y
3
4
1
y2
x 4 y 1 z 8
f.) 4  5
x y z
Example 2: The population of Pennsylvania is about 1.24 × 107. The population of North Dakota is
about 6.4 × 105. About how many more times greater is the population of Pennsylvania than North
Dakota?
Example 3: Population density describes the number of people per unit area. During one year, the
population of Angola was 1.21 × 107 people. The area of Angola is 4.81 × 105 mi2. What was the
population density of Angola that year?
Example 4: During one year, the population of Honduras was 7.33 × 106 people. The area of
Honduras is 4.33 × 104 mi2. What was the population density of Honduras that year?
3
x
How would you write   ?
 y
EXPONENT PROPERTY:
RAISING A QUOTIENT TO A POWER:
To raise a quotient to a power, raise the numerator and denominator to the power and simplify.
n
an
a
   n , where a ≠ 0, b ≠ 0, and n are rational numbers.
b
b
3
5
1
2
 x
x
   5
y
 y
27
3 3
Examples:    3 
 5  5 125
3
1
2
a
a
   1
b
b2
5
Examples 5a – 5d: Simplify each expression.
 23
z
a.) 
 5






3
 4
b.)  3 
x 
a
How would you write  
b
 34
a
c.)  5
a

2





4
 w5 
d.)  
 4 
3
n
?
Examples 6a – 6d: Simplify each expression.
 2 x6 
a.)  4 
 y 
 3c 3 
c.)  2 
d 
3
 a 
b.)  
 5b 
4
2
 2b 4 
d.)  3 
 c 
3
Review of Properties:
Examples 7a – 7i: Simplify each expression
a.) 7 x  2 x
3
4

3
b.) 2 x y

2 3
8x 4
c.)
4x 2
 2x 

d.) 
 4y 
2

e.) 3x 2 y

2
 8x 3 
f.)  5 
 4x 
2
 5x 2 

g.) 
3 
 10 x 
2
 4x3 

h.) 
5 
 12 x 
2
 7 y 3   12 x 4 
   2 
 14 x   9 y 
i.) 