Download Lecture notes for Section 7.3

Int. Alg. Notes
Section 7.3
Page 1 of 4
Section 7.3: Simplifying Radical Expressions
Big Idea: A radical is in simplest form when:
1. As many powers as possible are pulled out of the radical (i.e., the radicand does not contain any factors
that are perfect powers of the index).
2. The index of the radical is as low as possible.
3. There are no radicals in the denominator.
Big Skill: You should be able to simplify radical expressions using the criteria above.
Vocabulary:
1. Radicand is the expression under the radical.
2. The index (or order) of a radical is the number indicating the root being taken.
The Laws of Exponents (from section 7.2):
a  1 for a  0
Power Rule:
0
Zero Exponent Rule:
1
for a  0
an
a m  a n  a mn
a n 
Negative Exponent Rule:
Product Rule:
Quotient Rule:
a
a
1
2
1
n
a 
m n
 a mn
Product to Power Rule:
ab n
Quotient to Power Rule:
an
a
   n
b
b
 a nb n
n
am
1
 a mn  nm
n
a
a
a
Quotient to a Negative Power Rule:  
b
Meaning of Rational Exponents (from section 7.1):
 a
a
Quotient Property of Radicals:
3
3a
n
m
a n  n am  n a
Properties of Radicals (new for section 7.3):
Simplifying the nth root of a perfect nth power:
a n b  n ab
  
n
b
 
a
 
m
n a
Product Property of Radicals:
1
n
n
a
n
b
n
n
a n  a if n  3 is odd
n
a n  a if n  2 is even
a
b
Product Property of Radicals:
If n a and n b are real numbers, and n  2 is an integer, then
 a  b  
n
n
n
ab .
Steps to simplify a radical expression:
1. Write each factor of the radicand as the product of two factors, one of which is a perfect power of the
radicand.
2. Write the radicand as the product of two radicals (using the product property of radicals).
3. Take the nth root of the perfect power.
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 7.3
Page 2 of 4
Practice: “Pull” perfect powers of the index out of the radical using the product property of radicals:
1. 150 
2.
4
75z 2 
3.
4.
5.
48 
3
16a2b3c5d 7 
6  63

3
Practice: Use the product property to combine then simplify.
6. 11 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
7.
3
Section 7.3
Page 3 of 4
18 3 6 
8. 2 3 9 x2 3 3x2 
9.
4
9m3n5 4 12m2n5 
Practice: Use the quotient property to combine then simplify.
24
10.

49
11.
12.
3
5p

27
45 y 5
5y

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
13.
2 3 250n
3
2n 4
Section 7.3
Page 4 of 4

Practice: Multiply the following radicals with unlike indices by first re-writing the radicals using rational
exponents, and then simplifying.
14. 2 3 2 
15.
3
4 4 25 
16.
6
4 9 48 
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.