Transcript Section 2.2 – Properties of Exponents

2.2 Properties of Exponents
• Objectives:
– Evaluate expressions involving exponents.
– Simplify expressions involving exponents.
• Standard:
– 2.1.11.A. Use operations (e.g. raising to a
power).
The expression an is called a power of a.
In the expression, a is called the base and n the
exponent.
• I. Definition of Integer Exponents
• Let a be a real #.
• If n is a natural #, then an = a * a * a . . * a, n times.
• If a is nonzero, then a0 = 1.
• If n is a natural #, then
a
n
1
 n
a
Properties of Exponents
Let a and b be nonzero real numbers. Let m and n be integers
Product of Powers
a  a 
Quotient of Powers
am
mn
a
n
a
m
n
a
Power of a Power
a 
a
Power of a Product
ab 
 a nb n
m n
n
n
Power of a Quotient
mn
n
a
a
 
   n
b
b
m n
Examples:


1. Simplify 3x 2 y 2  2 x 3 y 4 . Write your answer wit h positive exponents only. *
 

2. Simplify 2 z 3x 2 5 z 3 . Write your answers with positive exponents only.
*
4
 y 
3. Simplify  12 3  . Write your answer wit h positive exponents only.
 2z y 
7


2
4. Simplify - 2a 5b 3 . Write your answer wit h positive exponents only.
 -3b c
5. Simplify  2 7
 cb
2 5
3

 . Write your answer wit h positive exponents only.


6. Simplify x -3 y -1
 x
1
3

2
y 0 . Write your answer wit h positive exponents only.
3
12
3
 3z   3x yz 
 . Write your answer wit h positive exponents only.
7. Simplify   4  
7
 x   2 xy 
2
Rational Exponents
An expression with rational exponents can be represente d in an equivalent
form that involves the radical symbol,
For example, a
1
3
3
.
1
equals a because when a 3 is cubed, the result is a, as
as shown in your notes. This is the definition of 3 a .
Definition of Rational Exponents
For all positive real numbers a :
1
If n is a nonzero integer, a n  n a . If m and n are integers and n  0,
m
1
a n  (a n )m  (n a )m
Example
1
a. 16
b. 27
4
c. 125
d. 8
2
f. 36
3
1
3
e. 64
3
81
5
4
1
3
2
4
3
2
4
216
Writing Activities:
1). Describe how to use Properties of
Exponents to simplify 307 36.
1/ 2
2). Explain why 25 and -5 are not
equivalent.
Homework
Integrated Algebra II: Section 2.2 Level A
Honors Algebra II: Section 2.2 Level B
PSSA WARM-UP QUESTION 
Algebra II - Chp. 2
Standard 2.1.11 A Find Powers.
• What are some of the properties of exponents?
Properties of Exponents
Multiplication of Exponents
Power to a Power
Division of Exponents
Raised to Zero
Negative Exponent
Rule
Example
Example 1
• The equation for centripetal acceleration is given as:
Ac  4  2 RT 2
1b. Find the centripetal acceleration in ft. per second squared of a rider
who makes one rotation in 5 seconds and whose radius is 6 feet.
r = 6, T = 5
Ac = 9.47 feet per second squared