Download Int Alg Lecture Notes, Section 7.2

Int. Alg. Notes
Section 7.2
Page 1 of 4
Section 7.2: Simplifying Expressions Using the Laws of Exponents
Big Idea: Integer exponents represent repeated multiplication. As such, there are formulas for simplifying
expressions with exponents whose basis lie in the concept of repeated multiplication or cancellation. These
formulas extend to rational exponents (and real-valued exponents) as well.
Big Skill: You should be able to simplify expressions using these laws of exponents.
The Laws of Exponents:
The following formulas apply when a and b are any real numbers, and m and n are rational numbers.
Law
Example
0
Divide both sides of the following equation by 2 five
Zero Exponent Rule:
a  1 for a  0
times to see one reason why zero and negative
1
a  n  n for a  0 
Negative Exponent Rule:
exponents are defined as they are:
a
23  8





Product Rule:
a m  a n  a mn
23  24 
Quotient Rule:
am
1
 a mn  nm
n
a
a
36

32
Power Rule:
a 
 a mn
5 
Product to Power Rule:
ab n
 a nb n
3  7
Quotient to Power Rule:
an
a

 
bn
b
m n
2 3
n
a
Quotient to a Negative Power Rule:  
b

3

4
 2
  
 3
n
b
 
a
n
3
 3
  
5
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 7.2
Page 2 of 4
An expression with rational exponents is considered simplified when:
1. All the exponents are positive.
2. Each base occurs only once.
3. There are no parentheses in the expression (unless there is addition or subtraction of a base involved).
4. There are no powers written to powers.
5. Rational exponents are reduced as much as possible (this implies that the radicals are of the smallest
index possible, and that the base of a rational exponent is as small as possible).
Practice:
2
3
1. Simplify 16 16
5
6
2
2. Simplify
43
4

5
6
5
 32  3
3. Simplify  7 
 
  32 14 
4. Simplify  a b 


8
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 7.2
Page 3 of 4
4
  43 2  2 12  3
5. Simplify  x y  x y 



 1 52 
 2x y 
6. Simplify  2 2 
x y
7. Simplify
6
93
8. Simplify
3
27a3b9
4
x3
x
9. Simplify
5
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
10. Simplify
Section 7.2
3
Page 4 of 4
n
3
1
1
1
3

11. Simplify 3x 2  2 x 2  4 x  1 by factoring out x 2 .
2
3
12. Simplify 2 x  x

 3x  2 
1
3
by factoring out x .
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.